similar argument then shows that $w\in
X_{s(w_{n(w)})}^{-}w_{n(w)}x^{s-max}_{r(w_{n(w)})}\subseteq[y,z]$. We conclude
that

$[y,z]\subseteq\bigcup_{e\in E_{0}}X_{s(e)}^{-}ex^{s-max}_{r(e)}.$

Moreover, the sets on the right are clearly pairwise disjoint. As each is an
interval, they are linearly order by $\leq_{r}$. Let us transfer this linear
order $E_{0}$ in the obvious way:
$E_{0}=\\{e_{1}=y_{n(y)},e_{2},\ldots,e_{K}=z_{n(z)}\\}$.

Let $1\leq k<K$ be in $E_{0}$ and consider
$w_{k}=x_{s(e_{k})}^{r-max}e_{k}x^{s-max}_{r(e_{k})}$, which is clearly in
$[y,z)$. Hence, its $r$-successor, $\Delta_{r}(w)$ is also in $[y,z]$ and
hence lies in
$X_{s(e_{k^{\prime}})}^{-}e_{k^{\prime}}x^{s-max}_{r(k^{\prime})}$, for some
$k^{\prime}$. As there are no points strictly between $w$ and $\Delta_{r}(w)$,
we must have $k^{\prime}=k+1$.

Let $f_{k}=S_{s}(e_{k}),1\leq k\leq K$. From part 1, we know that, for each
$k$,

$\Delta_{s}(X_{s(e_{k})}^{-}e_{k}x^{s-max}_{r(e_{k})})=X_{s(f_{k})}^{-}f_{k}x^{s-min}_{r(f_{k})}.$

and its $r$-maximal element is $\Delta_{s}(w_{k})$. As $[y,z]$ is disjoint
from $\Sigma_{\mathcal{B}}$, for $1\leq k<K$, we have

$\Delta_{r}(\Delta_{s}(w_{k}))=\Delta_{s}(\Delta_{r}(w_{k}))\in\Delta_{s}(X_{s(e_{k+1})}^{-}e_{k+1}x^{s-max}_{r(e_{k+1})})=X_{s(f_{k+1})}^{-}f_{k+1}x^{s-min}_{r(f_{k+1})}.$

In particular, this means that the sets
$\Delta_{s}(X_{s(e_{k})}^{-}e_{k}x^{s-max}_{r(e_{k})})$ are all comparable in
$\leq_{r}$ and hence in the same $T^{+}$-class. It also follows that there are
no points between $X_{s(f_{k})}^{-}f_{k}x^{s-min}_{r(f_{k})}$ and
$X_{s(f_{k+1})}^{-}f_{k+1}x^{s-min}_{r(f_{k+1})}$. So the union
$\bigcup_{k}X_{s(f_{k})}^{-}f_{k}x^{s-min}_{r(f_{k})}$ is also an interval and
the map $\Delta_{s}$ preserves $\leq_{r}$ on $[y,z]$. ∎

######  Theorem 7.12.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram satisfying the
standard hypotheses. For any $y$ in $Y_{\mathcal{B}}$, the space
$\varphi_{r}^{y}(T^{+}(y)\cap Y_{\mathcal{B}})$ consists of a countable number
of open intervals, each is contained in a single equivalence class of
$T^{\sharp}(S_{\mathcal{B}})$. Moreover, the number of such intervals is

  1. (1)

$1+\\#\left(T^{+}(y)\cap X_{\mathcal{B}}^{s-min}\cap
X_{\mathcal{B}}^{s-min}-X_{\mathcal{B}}^{r-max}-X_{\mathcal{B}}^{r-min}\right)$,
if this is greater than $1$.

  2. (2)

$1+2^{-1}\\#T^{+}(y)\cap\Sigma_{\mathcal{B}}$, if this is greater than $1$.

  3. (3)

$1$.

The three cases are mutually exclusive.

If $\Sigma_{\mathcal{B}}$ is infinite, then this number is infinite, for some
$y$ in $Y_{\mathcal{B}}$.

######  Corollary 7.13.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram satisfying the
standard hypotheses and let $y$ be in $Y_{\mathcal{B}}$. $T^{\sharp}(\pi(y))$
has a connected component which is closed in $S_{\mathcal{B}}$ if and only if

  1. (1)

$T^{+}(y)\cap(X_{\mathcal{B}}^{r-max}\cup X_{\mathcal{B}}^{r-min})$ is non-
empty and the number of connected components of $\varphi^{y}_{r}(T^{+}(y)\cap
Y_{\mathcal{B}})$ is at least $2$ or

  2. (2)

$T^{+}(y)\cap(X_{\mathcal{B}}^{r-max}\cup X_{\mathcal{B}}^{r-max})$ is empty
and the number of connected components of $\varphi^{y}_{r}(T^{+}(y)\cap
Y_{\mathcal{B}})$ is at least $3$.

In all other cases, each connected component of $T^{\sharp}(\pi(y))$ is dense
in $S_{\mathcal{B}}$.

######  Definition 7.14.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram satisfying the
hypotheses of 6.2. We define $\mathcal{F}_{\mathcal{B}}^{+}$ to be the open
subgroupoid of $T^{\sharp}(S_{\mathcal{B}})$ consisting of those pairs which
lie in the same connected component of their equivalence class. For any $x$ in
$S_{\mathcal{B}}$, we denote its equivalence class in
$\mathcal{F}_{\mathcal{B}}^{+}$ by $\mathcal{F}^{+}_{\mathcal{B}}(x)$.

Although we will not need the following result immediately, it will be useful
to record it now.

######  Proposition 7.15.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram satisfying the
hypotheses of 6.2 and let $m<n$.

  1. (1)

If $p$ is any path in $E^{Y}_{m,n}$ and $x$ is any element of $X^{+}_{r(p)}$
which does not consist of all s-maximal edges or s-minimal edges, then
$\pi(X^{-}_{s(p)}px)$ is contained in $\mathcal{F}^{+}_{\mathcal{B}}(\pi(x))$.

  2. (2)

If $p=(p_{1},p_{2})$ is any element of $E^{s}_{m,n}$, then
$\pi(X^{-}p_{1}x_{r(p_{1})}^{s-max})=\pi(X^{-}p_{2}x_{r(p_{2})}^{s-min})$ is
contained in a single element of $\mathcal{I}_{\mathcal{B}}$, which we denote
by $I(p)$.

  3. (3)

If $=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ are any elements in $E^{s}_{m,n}$
with $r(p)=r(q)$, then

$\pi\times\pi\left(X^{-}_{s(p_{1})}p_{1}x^{s-max}_{r(p_{1})}\times
X^{-}_{s(q_{1})}q_{1}x^{s-max}_{r(q_{1})}\right)\subseteq\mathcal{F}_{\mathcal{B}}^{+}$

if and only if $I(p)=I(q)$.

###### Proof.

For the first part, the hypothesis that $p$ is in $E_{m,n}^{Y}$ ensures that
$X^{-}_{s(p)}px$ is contained in $Y_{\mathcal{B}}$. Moreover, by part 4 of
Lemma 4.4, it is a compact interval in the order $\leq_{r}$ on $T^{+}(x)$. It
follows from Lemma 4.9 that its image under $\pi$ is also a compact interval
in a set which is homeomorphic to $\mathbb{R}$ and hence is connected.

The second part is done in the same way. The third part follows from the
definitions of $I(p),I(q)$ given in part 2 and $\mathcal{F}_{\mathcal{B}}$. ∎

The following is a fairly easy consequence of the information we have about
the connected components of $T^{\sharp}(S_{\mathcal{B}})$. While the
hypotheses seem somewhat restrictive, we will see that they hold in the case
of the typical finite genus surfaces and the Bratteli diagrams given by Rauzy-
Veech induction.

######  Theorem 7.16.

Let $\mathcal{B}$ be a bi-infinite Bratteli diagram satisfying our standard
hypotheses. If $X^{s-min}_{\mathcal{B}}\cup X^{s-max}_{\mathcal{B}}\subseteq
X^{r-max}_{\mathcal{B}}\cup X^{r-min}_{\mathcal{B}}$ and
$\Sigma_{\mathcal{B}}$ is empty, then
$T^{\sharp}(S_{\mathcal{B}})=\mathcal{F}_{\mathcal{B}}^{+}$.

######  Theorem 7.17.

Let $\mathcal{B}$ be a bi-infinite Bratteli diagram satisfying our standard
hypotheses. The foliation $\mathcal{F}_{\mathcal{B}}^{+}$ is minimal if and
only if

  1. (1)

For each $x$ in $X^{r-max}_{\mathcal{B}}\cup X^{r-min}_{\mathcal{B}}$,
$T^{+}(x)\cap\Sigma_{\mathcal{B}}$ is empty and

  2. (2)

For each $x$ with $T^{+}(x)\cap X^{r-max}_{\mathcal{B}}\cup
X^{r-min}_{\mathcal{B}}$ empty, we have

$\\#T^{+}(x)\cap\left(X^{s-min}_{\mathcal{B}}\cup
X^{s-max}_{\mathcal{B}}\cup\Sigma_{\mathcal{B}}\right)\leq 1.$

In particular, if $\Sigma_{\mathcal{B}}$ is infinite, then
$\mathcal{F}_{\mathcal{B}}^{+}$ is not minimal.

##  8\. $C^{*}$-algebras

We now begin our investigations into the various $C^{*}$-algebras associated
with the groupoids of the last section.

The most obviously important ones are that associated with right (left) tail
equivalence on the Bratteli diagram (which are both an AF-algebra) and the
$C^{*}$-algebra associated with the horizontal (vertical, respectively)
foliation of the surface. Our goals are, first, to show that the latter is a
$C^{*}$-subalgebra of the former and, second, to relate their $K$-theories.
For the second, our key technical ingredient is the results of [Put21].

On the other hand, it is clear from the last section that we actually have a
number of other groupoids of interest and each has its own $C^{*}$-algebra. In
fact, there is a natural, though rather long, passage from the AF-algebra to
the foliation algebra, each step producing a $C^{*}$-algebra from a groupoid
of the previous section.

It will probably be helpful to start with a brief outline of the individual
steps. Let us begin with a bi-infinite ordered Bratteli diagram,
$\mathcal{B}$, satisfying some conditions which we do not specify for the
moment. The details of each step will be given in the following subsections.

  1. (1)

The AF-algebra which we denote by $A_{\mathcal{B}}^{+}$ is the groupoid
$C^{*}$-algebra associated with tail equivalence on the space
$X_{\mathcal{B}}$, $T^{+}(X_{\mathcal{B}})$. We provide a canonical inductive
system of finite-dimensional subalgebras.

  2. (2)

Our first step is to simply restrict the equivalence relation
$T^{+}(X_{\mathcal{B}})$ to the open set $Y_{\mathcal{B}}\subseteq
X_{\mathcal{B}}$, which we are denoting by $T^{+}(Y_{\mathcal{B}})$, and we
let $A^{Y+}_{\mathcal{B}}$ be its $C^{*}$-algebra. It is naturally a full,
hereditary subalgebra of $A_{\mathcal{B}}^{+}$. It is also an AF-algebra and
the inclusion induces an isomorphism on $K$-theory. Again, we provide a
canonical inductive system of finite-dimensional subalgebras. In short, the
passage from $A_{\mathcal{B}}^{+}$ to $A^{Y+}_{\mathcal{B}}$ is rather minor.

  3. (3)

The next step is to pass to the space $S^{s}_{\mathcal{B}}$ with its
equivalence relation $T^{\sharp}(S^{s}_{\mathcal{B}})$ and we will denote its
$C^{*}$-algebra by $B_{\mathcal{B}}^{+}$.

From part 2 of Definition 6.3, we see that the $S^{s}_{\mathcal{B}}$ is a
quotient of $Y_{\mathcal{B}}$. The quotient map does not respect the groupoids
$T^{+}(Y_{\mathcal{B}})$ and $T^{\sharp}(S^{s}_{\mathcal{B}})$, but from
Definition 7.5, $T^{+}(Y_{\mathcal{B}})$ contains the open subgroupoid
$T^{\sharp}(Y_{\mathcal{B}})$ and the inclusion induces an inclusion of
$C^{*}$-algebras, $C^{*}(T^{\sharp}(Y_{\mathcal{B}}))\subseteq
C^{*}(T^{+}(Y_{\mathcal{B}}))$.

The factor map $\pi^{s}$ then does induce an inclusion

$B_{\mathcal{B}}^{+}=C^{*}(T^{\sharp}(S^{s}_{\mathcal{B}}))\subseteq
C^{*}(T^{\sharp}(Y_{\mathcal{B}}))\subseteq
C^{*}(T^{+}(Y_{\mathcal{B}}))=A^{Y+}_{\mathcal{B}}.$

While $A^{Y+}_{\mathcal{B}}$ is an AF-algebra and we have given an explicit
sequence of finite-dimensional approximating subalgebras, the
$C^{*}$-subalgebra $B_{\mathcal{B}}^{+}$ has trivial intersection with these
finite-dimensional subalgebras. However, we give another inductive system for
$A^{Y+}_{\mathcal{B}}$ and, although the elements are not finite dimensional,
their intersections with $B_{\mathcal{B}}^{+}$ provide a natural inductive
system for the latter.

  4. (4)

In the other direction, we may pass to the space $S^{r}_{\mathcal{B}}$ with
its equivalence relation $T^{+}(S^{r}_{\mathcal{B}})$.

From part of Definition 6.3, we see that the $S^{r}_{\mathcal{B}}$ is also a
quotient of $Y_{\mathcal{B}}$. Unlike the last case, the quotient map only
acts non-trivially within the equivalence classes and, while it is
topologically non-trivial, at the level of measure theory it induces an
isomorphism. We make all of that precise and prove as a consequence that the
map on groupoids induces an isomorphism between
$C^{*}(T^{+}(S^{r}_{\mathcal{B}}))$ and $C^{*}(T^{+}(Y_{\mathcal{B}}))$.

  5. (5)

The next step is to consider $S_{\mathcal{B}}$ and the equivalence relation
$T^{\sharp}(S_{\mathcal{B}})$. Again from part 2 of Definition 6.3, we see
that this is a quotient of $T^{\sharp}(S^{s}_{\mathcal{B}})$. However, here
the only change is within individual equivalence classes. Moreover, within
each equivalence class, the quotient map induces an isomorphism of measure
spaces and the consequence is that
$C^{*}(T^{\sharp}(S_{\mathcal{B}}))=C^{*}(T^{\sharp}(S^{s}_{\mathcal{B}}))$.
In other words, this step has no real effect at all.

  6. (6)

The final step is to go from the $C^{*}$-algebra
$B_{\mathcal{B}}^{+}=C^{*}(T^{\sharp}(S_{\mathcal{B}}))$ to the foliation
$C^{*}$-algebra, $C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$. Here again, we will
exhibit a similar inductive limit structure for the latter.

###  8.1. AF-algebras

######  Definition 8.1.

Let $\mathcal{B}$ be a bi-infinite Bratteli diagram. The $C^{*}$-algebra
$A_{\mathcal{B}}^{+}$ is defined to be the groupoid $C^{*}$-algebra of
$T^{+}(X_{\mathcal{B}})$ with Haar system $\nu_{r}^{x},x\in X_{\mathcal{B}}$.

It is probably worth noting that this does not need an _ordered_ Bratteli
diagram. Also, it uses a state, but it is independent of the choice.

######  Proposition 8.2.

Let $m<n$ be integers and let $p,q$ be in $E_{m,n}$ with $r(p)=r(q)$. For
$(x,y)$ in $T^{+}(X_{\mathcal{B}})$, define

$a_{p,q}(x,y)=\nu_{r}(s(p))^{-1/2}\nu_{r}(s(q))^{-1/2}$

if $(x_{m+1},\ldots,x_{n})=p,(y_{m+1},\ldots,y_{n})=q$ and $x_{i}=y_{i}$, for
all $i>n$. Define $a_{p,q}(x,y)=0$ otherwise. Then $a_{p,q}$ is a continuous,
compactly supported function om $T^{+}(Y_{\mathcal{B}})$. Moreover, we have

  1. (1)

If $p^{\prime},q^{\prime}$ is another pair in $E_{m,n}$ with
$r(p^{\prime})=r(q^{\prime})$, then

$a_{p,q}a_{p^{\prime},q^{\prime}}=\left\\{\begin{array}[]{cl}a_{p,q^{\prime}}&\text{
if }q=p^{\prime},\\\ 0&\text{ if }q\neq p^{\prime}\\\ \end{array}\right.$

In particular, if $r(p)\neq r(p^{\prime})$ then this product is zero.

  2. (2)

$a_{p,q}^{*}=a_{q,p}$.

  3. (3)

$a_{p,q}=\sum_{s(e)=r(p)}a_{pe,qe}$,

  4. (4)

$a_{p,q}=\sum\nu_{r}(s(e))^{-1/2}\nu_{r}(s(f))^{-1/2}\nu_{r}(r(e))^{1/2}\nu_{r}(r(f))^{1/2}a_{ep,fq}$,
where the sum is over all $e,f$ in $E_{m-1}$ with $r(e)=s(p),r(f)=s(q)$.

######  Proposition 8.3.

For integers $m<n$, let $A_{m,n}^{+}$ denote the span of all elements
$a_{p,q}$, where $p,q$ are in $E_{m,n}$ with $r(p)=r(q)$. If $v$ is a vertex
in $V_{n}$, let $A_{m,n,v}^{+}$ denote the span of all elements $a_{p,q}$,
where $p,q$ are in $E_{m,n}$ with $r(p)=r(q)=v$.

  1. (1)

$A_{m,n,v}^{+}$ is isomorphic to $M_{j}(\mathbb{C})$, where $j$ is the number
of paths $p$ in $E_{m,n}$ with $r(p)=v$.

  2. (2)

$A_{m,n}^{+}=\oplus_{v\in V_{n}}A_{m,n,v}^{+}$, In particular, each
$A_{m,n}^{+}$ is a finite dimensional $C^{*}$-subalgebra of
$A^{+}_{\mathcal{B}}$.

  3. (3)

For all $m,n$, $K_{0}(A_{m,n}^{+})\cong\mathbb{Z}^{\\#V_{n}}$.

  4. (4)

For all $m,n$ we have $A_{m-1,n}^{+}\subseteq A_{m,n}^{+}\subseteq
A_{m,n+1}^{+}$.

  5. (5)

With the identifications above, the inclusion $A_{m-1,n}^{+}\subseteq
A_{m,n}^{+}$ is the identity map on $K_{0}$ and the inclusion
$A_{m,n}^{+}\subseteq A_{m,n+1}^{+}$ is the map on $K_{0}$ given by the edge
matrix for $E_{n+1}$.

  6. (6)

The union of $A_{m,n}^{+}$ over all $m,n$ is dense in $A^{+}_{\mathcal{B}}$.

It will be useful for us to identify another sequence of approximating
subalgebras, although these are not finite-dimensional.

######  Proposition 8.4.

For each $m<n$ and vertex $v$ in $V_{n}$, define $AC_{m,n,v}^{+}$ to be the
closure on the span of all $a_{pp^{\prime},qp^{\prime}}$, where $p,q$ are in
$E_{m,n}$ with $r(p)=r(q)=v$, and $p^{\prime}$ is in $E_{n,n^{\prime}}$ with
$n\leq n^{\prime}$ and $s(p^{\prime})=v$.

The map which sends $a_{pp^{\prime},qp^{\prime}}$ to
$a_{p,q}\otimes\chi_{p^{\prime}X_{r(p^{\prime})}^{+}}$ extends to an
isomorphism from $AC_{m,n,v}^{+}$ to $A_{m,n,v}^{+}\otimes C(X_{v}^{+})$.

We also let $AC_{m,n}^{+}$ be the span of all $AC_{m,n,v}^{+},v\in V_{n}$ and
we have
$AC_{m,n}^{+}=\oplus_{v}AC_{m,n,v}^{+}\cong\oplus_{n}A_{m,n,v}^{+}\otimes
C(X_{v}^{+})$.

There are a couple of different notations we may use for the elements of
$AC_{m,n}^{+}$. The first option is $(f_{v})_{v\in V_{n}}$, where
$f_{v}:X_{v}^{+}\rightarrow A_{m,n,v}^{+}$ is continuous. Alternately, we let
$f$ denote the union of the functions $f_{v},v\in V_{n}$ which we regard as a
function from $\cup_{v\in V_{n}}X_{v}^{+}$ to $A_{m,n}^{+}$ such that $f(x)$
is in $A_{m,n,v}^{+}$ if $x$ is in $X_{v}^{+}$.

Finally, for any $p,q$ in $E_{m,n},m<n$, with $r(p)=r(q)$, we let
$\tilde{a}_{p,q}$ be a continuous function from $X_{r(p)}^{+}$ to the one-
dimensional subspace of $A_{m,n}^{+}$, $\mathbb{C}a_{p,q}$. Every element of
$AC_{m,n}^{+}$ is a finite sum of such functions, over possible values of
$p,q$. Given such a finite sum, let us just record that we regard it as a
function on $T^{+}(X_{\mathcal{B}})$ by

$\left(\sum_{p,q}\tilde{a}_{p,q}\right)(x,y)=\tilde{a}_{x_{(m,n]},y_{(m,n]}}(x_{(n,\infty)})$

if $x_{(n,\infty)}=y_{(n,\infty)}$ and is zero otherwise, for $(x,y)$ in
$T^{+}(X_{\mathcal{B}})$.

It will be useful to note explicitly the inclusion of $AC_{m,n}^{+}$ in
$AC_{m,n^{\prime}}^{+}$ for $n^{\prime}>n$, as follows. If
$\tilde{a}_{p,q}:X_{r(p)}^{+}\rightarrow\mathbb{C}$ is continuous, then we may
write it as an element of $AC_{m,n^{\prime}}$ as
$\sum_{p^{\prime}}f_{p^{\prime}}$, where the sum is over all $p^{\prime}$ in
$E_{n,n^{\prime}}$ with $s(p^{\prime})=r(p)$ and $f_{p^{\prime}}$ is defined
on $X_{r(p^{\prime})}^{+}$ by
$f_{p^{\prime}}(x^{\prime})=\tilde{a}_{p,q}(p^{\prime}x^{\prime})a_{pp^{\prime},qp^{\prime}}$,
for $x^{\prime}$ in $X_{r(p^{\prime})}^{+}$.

###  8.2. Intermediate $C^{*}$-algebras

In this section, we want to describe various $C^{*}$-algebras associated with
some of the groupoids introduced earlier. Generally speaking, we will show
that these all sit as subalgebras of our AF-algebra, $A_{\mathcal{B}}^{+}$,
and all contain our foliation algebra, $C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$.

While the $C^{*}$-algebra $A_{\mathcal{B}}^{+}$ is defined for any bi-infinite
Bratteli diagram, the constructions of this section need an ordered bi-
infinite Bratteli diagram.

Our first step is to consider the restriction of the equivalence relation
$T^{+}(X_{\mathcal{B}})$ to the set $Y_{\mathcal{B}}$, which we denote by
$T^{+}(Y_{\mathcal{B}})$.

Recall that under standing hypotheses that, $X^{ext}_{\mathcal{B}}$ is finite,
$X^{S}_{\mathcal{B}}$ is countable and
$Y_{\mathcal{B}}=X_{\mathcal{B}}-X^{ext}_{\mathcal{B}}-X^{S}_{\mathcal{B}}$,
we have removed only set of measure zero in any measure in our Haar system and
so the collection of measures $\nu^{x}_{r},x\in Y_{\mathcal{B}},$ is also a
Haar system for the groupoid $T^{+}(Y_{\mathcal{B}})$.

######  Definition 8.5.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram. The
$C^{*}$-algebra $A^{Y+}_{\mathcal{B}}$ is defined to be the groupoid
$C^{*}$-algebra of $T^{+}(Y_{\mathcal{B}})$.

######  Proposition 8.6.

Under our standing hypotheses, $A^{Y+}_{\mathcal{B}}$ is a full hereditary
$C^{*}$-subalgebra of $A_{\mathcal{B}}^{+}$. In particular, the inclusion map
induces an isomorphism on $K$-theory.

The following is an immediate consequence of the definitions. We omit the
proof.

######  Proposition 8.7.

For $m<n$, if $p,q$ are in $E^{Y}_{m,n}$ with $r(p)=r(q)$, then $a_{p,q}$ is
in $A^{Y+}_{\mathcal{B}}$. We define $A^{Y+}_{m,n}$ to be the span of all such
elements. Assume that for every $v$ in $V_{m}$ and $w$ in $V_{n}$, there is a
path $p$ in $E_{m,n}^{Y}$ with $s(p)=v,r(p)=f$.

  1. (1)

The inclusion of $A^{Y+}_{m,n}$ in $A_{m,n}^{+}$ induces an order isomorphism
on $K$-theory.

  2. (2)

All conditions of 8.3 hold replacing $A_{m,n}^{+}$ with $A^{Y+}_{m,n}$ and
$A_{\mathcal{B}}^{+}$ with $A^{Y+}_{\mathcal{B}}$.

It will also be convenient for us to adopt the notation $AC^{Y+}_{m,n}$ to be
the elements of $AC_{m,n}^{+}$ of the form $\sum_{p,q\in
E_{m,n}^{Y}}\tilde{a}_{p,q}$; that is, functions taking values in
$A^{Y+}_{\mathcal{B}}$.

Our next aim is to analyze the $C^{*}$-algebra of the equivalence relation
$T^{\sharp}(S^{s}_{\mathcal{B}})$ on the space $S^{s}_{\mathcal{B}}$.

Recall from 6.3 that $S^{s}_{\mathcal{B}}$ is a quotient of $Y_{\mathcal{B}}$.
Moreover, we know from Proposition 7.7 that the quotient map respects the
equivalence relations $T^{\sharp}(Y_{\mathcal{B}})$ and
$T^{\sharp}(S^{s}_{\mathcal{B}})$.

Recall that $E^{s}_{m,n},m<n$ consists of pairs $p=(p_{1},p_{2})$ such that
$p_{1},p_{2}$ are in $E_{m,n}^{Y}$ and $p_{2}$ is the s-successor of $p_{1}$.
For $p=(p_{1},p_{2})$ in $E^{s}_{m,n}$, we define
$r(p)=(r(p_{1}),r(p_{2})),s(p)=(s(p_{1}),s(p_{2}))$. We also define

$G_{m,n}:\\{(p,q)\mid p,q\in E_{m,n}^{s},r(p)=r(q)\\},$

which is a finite equivalence relation and hence also a groupoid. For $i=1,2$,
we define $\alpha_{i}:G_{m,n}\rightarrow E^{Y}_{m,n}\times E^{Y}_{m,n}$ by
$\alpha_{i}((p_{1},p_{2}),(q_{1},q_{2}))=(p_{i},q_{i})$. It should cause no
confusion if we also define $\alpha_{i}:C^{*}(G_{m,n})\rightarrow A^{Y}_{m,n}$
by $\alpha_{i}(g)=\sum_{(p,q)\in G_{m,n}}g(p,q)a_{\alpha_{i}(p,q)}$, for any
function $g:G_{m,n}\rightarrow\mathbb{C}$. It is a simple matter to verify
that $\alpha_{1},\alpha_{2}$ are $*$-homomorphisms.

######  Proposition 8.8.

Let $\mathcal{B}$ be an ordered bi-infinite Bratteli diagram satisfying the
conditions of 6.2. For each $m<n$, suppose that, for $(x,y)$ in
$T^{+}(X_{\mathcal{B}})$, $\tilde{a}=\sum_{p,q\in E_{m,n}}\tilde{a}_{p,q}$
with $\tilde{a}_{p,q}:X_{r(p)}^{+}\rightarrow\mathbb{C}a_{p,q}$ in
$AC^{Y+}_{m,n}$. Then $\tilde{a}$ is in $B_{\mathcal{B}}^{+}$ if and only if
the following hold:

  1. (1)

for any $p,q$ in $E_{m,n}^{Y}$ with $r(p)=r(q)$,
$\tilde{a}_{p,q}=\tilde{b}_{p,q}\circ\varphi_{s}^{r(p)}$, where
$\tilde{b}_{p,q}:[0,\nu_{s}(r(p))]\rightarrow\mathbb{C}$ is continuous and
$\varphi^{r(p)}_{s}$ is as in 4.7,

  2. (2)

for every $(p,q)$ in $E^{s}_{m,n}$ with $r(p)=r(q)$, we have

$\tilde{a}_{p_{1},q_{1}}(x_{r(p_{1})}^{s-max})=\tilde{a}_{p_{2},q_{2}}(x_{r(p_{2})}^{s-min}).$

  3. (3)

if $(p,q)$ is not in $\alpha_{1}(G_{m,n})$, then
$\tilde{a}_{p,q}(x_{r(p)}^{s-max})=0$,

  4. (4)

if $(p,q)$ is not in $\alpha_{2}(G_{m,n})$, then
$\tilde{a}_{p,q}(x_{r(p)}^{s-min})=0$,

We define $B_{m,n}^{+}$ to be the set of all elements, $\tilde{a}$, satisfying
these conditions.

###### Proof.

We will first show that any element satisfying the conditions lies in
$B_{\mathcal{B}}^{+}$. Let us just recall from the last subsection that, for
$(x,y)$ in $T^{+}(X_{\mathcal{B}})$,
$\tilde{a}(x,y)=\tilde{a}_{x_{(m,n]},y_{(m,n]}}(x_{(n,\infty)})$ if
$x_{(n,\infty)}=y_{(n,\infty)}$ and is zero otherwise.

It suffices to show that, for any $(x,y)$ in $T^{+}(Y_{\mathcal{B}})$,
$\tilde{a}(x,y)$ is zero if $(x,y)$ is not in $T^{\sharp}(Y_{\mathcal{B}})$
and that $\tilde{a}(x,y)=\tilde{a}(\Delta_{s}(x),\Delta_{s}(y))$, if $x,y$ are
in $\partial^{s}X_{\mathcal{B}}$. It is clear that $T^{+}(Y_{\mathcal{B}})$
and $T^{\sharp}(Y_{\mathcal{B}})$ agree except on
$\partial^{s}X_{\mathcal{B}}$ and so for both conditions we need only consider
the cases when $x,y$ are in $\partial^{s}X_{\mathcal{B}}$. Without loss of
generality, assume that $x_{n}$ is s-maximal, for all $n$ sufficiently large.
Hence, $y_{n}$ is also.

We first observe that if $\tilde{a}(x,y)$ is non-zero, then both $x_{(m,n]}$
and $y_{(m,n]}$ are in $E^{Y}_{m,n}$, which implies they are both in
$Y_{\mathcal{B}}$. In addition, we must have $(x,y)$ in
$T^{+}_{n}(X_{\mathcal{B}})$.

We first consider the case when either $n(x)\leq m$ or $n(y)\leq m$. It
follows that $x_{(m,n]}$ or $y_{(m,n]}$ are not in $E^{Y}_{m,n}$. The second
case is when $n(x)>n$. As $(x,y)$ is in $T^{+}_{n}(X_{\mathcal{B}})$,
$n(y)=n(x)$ and both lie in $X_{r(x_{n})}^{+}$. Moreover, in this case, it is
clear that $(\Delta_{s}(x),\Delta_{s}(y))$ also lies in
$T^{+}_{n}(X_{\mathcal{B}})$. The first condition ensures
$\tilde{a}=\tilde{b}\circ\varphi_{s}^{r(p)}$ and hence by part 3 of Lemma 4.3,
$\tilde{a}(\Delta_{s}(x),\Delta_{s}(y))=\tilde{a}(x,y)$, as desired.

The case which remains is $m<n(x),n(y)\leq n$. If either $S_{s}(x_{(m,n]})$ or
$S_{s}(y_{(m,n]})$ is not in $E^{Y}_{m,n}$ then $\tilde{a}(x,y)=0$ by the
third condition. It is also clear that
$\tilde{a}(\Delta_{s}(x),\Delta_{s}(y))=0$ in this case.

Next, we suppose that $S_{s}(x_{(m,n]})$ and $S_{s}(y_{(m,n]})$ are in
$E^{Y}_{m,n}$, but
$r(S_{s}(x_{(m,n]}))\neq r(S_{s}(y_{(m,n]}))$. Again by the third condition
$\tilde{a}(x,y)=0$ while $\tilde{a}(\Delta_{s}(x),\Delta_{s}(y))=0$ since
$(\Delta_{s}(x),\Delta_{s}(y))$ is not in $T^{+}_{n}(Y_{\mathcal{B}})$.

We are left with the case that $S_{s}(x_{(m,n]})$ and $S_{s}(y_{(m,n]})$ are
in $E^{Y}_{m,n}$ and $r(S_{s}(x_{(m,n]}))=r(S_{s}(y_{(m,n]}))$. Here, the
second condition, using $p_{1}=p,p_{2}=S_{s}(p),q_{1}=q,q_{2}=S_{s}(q)$
clearly implies $\tilde{a}(\Delta_{s}(x),\Delta_{s}(y))=\tilde{a}(x,y)$.

The converse direction is relatively simple and we omit the details. ∎

######  Theorem 8.9.

  1. (1)

For all $m<n$, $B_{m,n}^{+}$ is a $C^{*}$-subalgebra of $B_{\mathcal{B}}^{+}$.

  2. (2)

For all $m<n$, $B_{m,n}^{+}\subseteq B_{m-1,n+1}^{+}$.

  3. (3)

$\bigcup_{n=1}^{\infty}B_{-n,n}^{+}$ is dense in $B_{\mathcal{B}}^{+}$.

  4. (4)

For all $m<n$, we have

$\displaystyle B_{m,n}^{+}$ $\displaystyle\cong$
$\displaystyle\\{\left((f_{v})_{v},g\right)\in\left(\oplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes C[0,\nu_{s}^{v}],C^{*}(G_{m,n})\right)\mid$
$\displaystyle\sum_{v}f(x_{v}^{s-max})=\alpha_{1}(g),\sum_{v}f(x_{v}^{s-min})=\alpha_{2}(g)\\}.$

While our $C^{*}$-algebras $B_{m,n}^{+}$ are not unital, the reader should
compare the result in part 4 with the definition of recursive subhomogeneous
$C^{*}$-algebras given in [Phi07].

######  Corollary 8.10.

For $m<n$, we have a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(G_{m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Our next step addresses comparisons between
$A^{Y+}_{\mathcal{B}}=C^{*}(T^{+}(Y_{\mathcal{B}}))$ and
$C^{*}(T^{+}(S^{r}_{\mathcal{B}}))$. In addition, we will relate
$B_{\mathcal{B}}^{+}=C^{*}(T^{+}(S^{s}_{\mathcal{B}}))$ and
$C^{*}(T^{+}(S_{\mathcal{B}}))$.

######  Theorem 8.11.

  1. (1)

Pre-composition of continuous functions of compact support with

$\pi^{r}\times\pi^{r}:T^{+}(Y_{\mathcal{B}})\rightarrow
T^{+}(S^{r}_{\mathcal{B}}),$

which is a continuous proper surjection, is a $*$-homomorphism from
$C_{c}(T^{+}(S^{r}_{\mathcal{B}}))$ to $C_{c}(T^{+}(Y_{\mathcal{B}}))$ and
extends to an isomorphism between $C^{*}_{r}(T^{+}(S^{r}_{\mathcal{B}}))$ and
$C^{*}_{r}(T^{+}(Y_{\mathcal{B}}))$.

  2. (2)

Pre-composition of continuous functions of compact support with

$\rho^{r}\times\rho^{r}:T^{\sharp}(S^{s}_{\mathcal{B}})\rightarrow
T^{\sharp}(S_{\mathcal{B}})$

which is a continuous proper surjection, is a $*$-homomorphism from
$C_{c}(T^{\sharp}(S_{\mathcal{B}}))$ to
$C_{c}(T^{\sharp}(S^{s}_{\mathcal{B}}))$ and extends to an isomorphism between
$C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}}))$ and
$C^{*}_{r}(T^{\sharp}(S^{s}_{\mathcal{B}}))$.

###### Proof.

We outline a proof of the first part. The fact that the map is continuous and
proper follows from Proposition 7.3. It is also an easy computation, using the
fact that each equivalence class is mapped to an equivalence class preserving
the Haar system, that it is a $*$-homomorphism. In addition, the map preserves
the reduced norm and so extends continuously to an isometry.

It remains to show that it is surjective, at the level of the
$C^{*}$-algebras. Let $(x,y)$ be in $T^{+}(Y_{\mathcal{B}})$. Choose $n$
sufficiently large so that $(x,y)$ is in $T^{+}(Y_{\mathcal{B}})$ and let
$m<n$. Let $p=x_{[m,n)}$ and $q=y_{[m,n)}$ and consider

$W=\\{(x^{\prime},y^{\prime})\mid x^{\prime}\in
U_{s(p)}^{-}pX_{r(p)}^{+},y^{\prime}\in
U_{s(q)}^{-}pX_{r(q)}^{+},x^{\prime}_{(n,\infty)}=y^{\prime}_{(n,\infty)}\\}$

This is clearly an open set in $T^{+}(Y_{\mathcal{B}})$ containing $(x,y)$.
Moreover, as $n,m$ vary these form a neighbourhood base at $(x,y)$. As $(x,y)$
vary these sets form a base for the topology of $T^{+}(Y_{\mathcal{B}})$. In
consequence, every continuous function of compact support in
$T^{+}(Y_{\mathcal{B}})$ can be written as a finite sum of functions supported
in sets as above.

We also observe that there is an obvious homeomorphism between $W$ and
$U_{s(p)}^{-}\times U_{s(q)}^{-}\times X_{r(p)}^{+}$

By the using the results of 6.6, the same is also true if we replace $W$ by
the set $\pi^{r}(W)$ in $T^{+}(S^{r}_{\mathcal{B}})$ which is homeomorphic to
$(0,\nu_{r}(s(p)))\times(0,\nu_{r}(s(q)))\times X_{r(p)}^{+}$. Under these
identifications, the map $\pi^{r}$ can be identified with
$\varphi^{p}_{r}\times\varphi^{q}_{r}\times id$. The key point in this is
that, due to part 5 4.3, $\varphi^{p}_{r}$ and the fact that the map is
injective on a set of full measure, it induces an isomorphism between the
measure space $(U_{s(p)}^{-},\nu_{r}^{p})$ and $(0,\nu_{r}(s(p)),\lambda)$. ∎

###  8.3. Foliation algebra

In this subsection, we turn our attention to the $C^{*}$-algebra of the
horizontal foliation, $C^{*}(\mathcal{F}_{\mathcal{B}})^{+}$. When it is
convenient, we will also denote $C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$ by
$C_{\mathcal{B}}^{+}$.

The first result is an immediate consequence of the fact that
$\mathcal{F}_{\mathcal{B}}^{+}$ is an open subgroupoid of
$T^{\sharp}(S_{\mathcal{B}})$.

######  Theorem 8.12.

$C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$ is a $C^{*}$-subalgebra of
$C^{*}(T^{\sharp}(S_{\mathcal{B}}))$.

We next want to show that $C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$ has an
inductive limit structure analogous to that of
$C^{*}(T^{\sharp}(S^{s}_{\mathcal{B}}))$ appearing in Theorem 8.9 and
Corollary 8.10. Recalling the definition of $G_{m,n}$ from subsection 8.2, we
begin by defining, for each $m<n$,

$H_{m,n}=\\{(p,q)\in G_{m,n}\mid I(p)=I(q)\\}.$

This is a subgroupoid of $G_{m,n}$.

We remark that an analogue of Proposition 8.8 holds: we simply change
$B_{\mathcal{B}}^{+}$ to $C_{\mathcal{B}}^{+}$ and replace $G_{m,n}$ in
conditions 3 and 4 by $H_{m,n}$. This is an immediate consequence of
Proposition 8.8 and part 3 of Proposition 7.15. We let $C_{m,n}^{+}$ be the
set of all elements satisfying these conditions; that is,
$C_{m,n}^{+}=AC^{Y+}_{m,n}\cap C_{\mathcal{B}}^{+}$.

We then obtain analogues of Theorem 8.9 and Corollary 8.10 which we state
precisely for the record.

######  Theorem 8.13.

  1. (1)

For all $m<n$, $C_{m,n}^{+}$ is a $C^{*}$-subalgebra of $C_{\mathcal{B}}^{+}$.

  2. (2)

For all $m<n$, $C_{m,n}^{+}\subseteq C_{m-1,n+1}^{+}$.

  3. (3)

$\bigcup_{n=1}^{\infty}C_{-n,n}$ is dense in $C_{\mathcal{B}}^{+}$.

  4. (4)

For all $m<n$, we have

$\displaystyle C_{m,n}^{+}$ $\displaystyle\cong$
$\displaystyle\\{\left((f_{v})_{v},h\right)\in\left(\oplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes C[0,\nu_{s}^{v}],C^{*}(H_{m,n})\right)\mid$
$\displaystyle\sum_{v}f(x_{v}^{s-max})=\alpha_{1}(h),\sum_{v}f(x_{v}^{s-min})=\alpha_{2}(h)\\}.$

######  Corollary 8.14.

For $m<n$, we have a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(H_{m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

##  9\. A Fredholm module

The aim of this section is to produce a Fredholm module for our
$C^{*}$-algebras. This will be crucial in the K-theory computations of the
next section.

The books by Blackadar [Bla86], Higson and Roe [HR01] and Connes [Con94] are
all good references for Fredholm modules. We remind readers that, for any
$C^{*}$-algebra $A$, a Fredholm module for $A$ consists of a Hilbert space
$\mathcal{H}$, a representation $\pi$ of $A$ on $\mathcal{H}$ and a bounded
operator $F$ on $\mathcal{H}$ such that $(F^{2}-1)\pi(a),(F-F^{*})\pi(a)$ and
$[\pi(a),F]=\pi(a)F-F\pi(a)$ are all compact operators, for each $a$ in $A$.
In our case, we will give the Hilbert space and representation of the AF-
algebra, $A_{\mathcal{B}}^{+}$. The operator $F$ will actually satisfy
$F^{2}=1,F=F^{*}$, but the crucial condition that $[\pi(a),F]$ is compact, for
each $a$, holds if we consider $a$ in the $C^{*}$-subalgebra
$A^{Y}_{\mathcal{B}}$. In fact, our Hilbert space comes with a natural
$\mathbb{Z}_{2}$-grading, the representation $\pi$ is by even operators, while
$F$ is odd. In other words, we will have an even Fredholm module.

The last discussion will probably not be very helpful to non-operator
theorists. Let us give a simple example where these properties will be clear.
At the same time, what is happening in the example is really exactly what is
going on in our situation to follow and so this should provide some intuition.

Let $X\subseteq[0,1]$ be the standard Cantor ternary set. Let us list the open
intervals in its complement (in $[0,1]$) as $(x_{n},y_{n}),n\geq 1$ (the order
is not important here). Let $\mathcal{H}$ be a Hilbert space with a canonical
basis indexed by the endpoints, $\delta_{x_{n}},\delta_{y_{n}}$. (One view is
to put an infinite measure on $X$ with point mass at each $x_{n}$ and $y_{n}$
and consider the space of square-integrable functions. The $C^{*}$-algebra of
continuous functions on $X$, $C(X)$ c an be represented as operators on this
Hilbert space by simple evaluation of the functions: we supress the
representation and simply write
$f\delta_{x_{n}}=f(x_{n})\delta_{x_{n}},f\delta_{y_{n})=f(y_{n}}\delta_{y_{n}}$,
for all $n\geq 1$.

Define an operator $F$ on this space Hilbert space by specifying
$F\delta_{x_{n}}=\delta_{y_{n}},F\delta_{y_{n}}=\delta_{x_{n}}$, for all
$n\geq 1$. It is trivial to see $F^{2}=I,F^{*}=F$. It is a simple matter to
check, if $f$ is locally constant, then $f(x_{n})=f(y_{n})$, for all but
finitely many $n$ and the operator $[F,f]=Ff-fF$ is finite rank. Only slightly
more subtle is that, for any $f$ in $C(X)$, $[F,f]$ is compact. Finally, if
$\pi:X\rightarrow[0,1]$ denotes the devil’s staircase, then $f$ in $C(X)$ has
the form $f=g\circ\pi$, for some $g$ in $C[0,1]$ if and only if $[F,f]=0$.

The construction of the Hilbert space and representation is a standard one
from the regular representation of the groupoids. Let us recall this to
establish some notation. Let $\mathcal{B}$ be a bi-infinite Bratteli diagram,
$X_{\mathcal{B}}$ be its path space, $T^{+}(X_{\mathcal{B}})$ be right-tail
equivalence and $\nu_{r}^{x},x\in X_{\mathcal{B}}$ be its Haar system. For
each $x$ in $X_{\mathcal{B}}$, we consider the Hilbert space
$L^{2}(T^{+}(x),\nu_{r}^{x})$ and the representation denoted $\lambda^{x}$ of
$C^{*}_{r}(T^{+}(X_{\mathcal{B}}))$ satisfying

$\left(\lambda^{x}(f)\xi\right)(y)=\int_{T^{+}(x)}f(y,z)\xi(z)\,d\nu_{r}^{x}(z),$

for any $f$ in $C_{c}(T^{+}(X_{\mathcal{B}})),\xi\in
L^{2}(T^{+}(x),\nu_{r}^{x})$ and $y$ in $T^{+}(x)$. We refer the reader to
Renault [Ren80] of Williams [Wil19] for further details.

######  Definition 9.1.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram. Let
$I_{\mathcal{B}}^{+},J_{\mathcal{B}}^{+}$ and $x_{i},1\leq i\leq
I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+},$ be as in 7.4. For $1\leq i\leq
I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}$, we define
$\mathcal{H}_{i}=L^{2}(T^{+}(x_{i}),\nu^{x_{i}}_{r}))$,

$\mathcal{H}^{max}_{\mathcal{B}}=\bigoplus_{1\leq
i\leq{I_{\mathcal{B}^{+}}}}\mathcal{H}_{i},\hskip
72.26999pt\mathcal{H}^{min}_{\mathcal{B}}=\bigoplus_{I_{\mathcal{B}^{+}}<i\leq
I_{\mathcal{B}^{+}}+J_{\mathcal{B}^{+}}}\mathcal{H}_{i},$

and
$\mathcal{H}_{\mathcal{B}}=\mathcal{H}^{max}_{\mathcal{B}}\oplus\mathcal{H}^{min}_{\mathcal{B}}$.
We define $\pi_{\mathcal{B}}=\displaystyle\bigoplus_{1\leq i\leq
I_{\mathcal{B}^{+}}+J_{\mathcal{B}^{+}}}\lambda^{x_{i}}$.

Finally, we define
$F_{\mathcal{B}}:\mathcal{H}_{\mathcal{B}}\rightarrow\mathcal{H}_{\mathcal{B}}$
to be the operator $(F_{\mathcal{B}}\xi)(x)=\xi(\Delta_{s}(x))$, for any $\xi$
in $\mathcal{H}_{\mathcal{B}}$ and $x$ in $\bigcup_{i}T^{+}(x_{i})$.

We make several observations. It would probably be more accurate to replace
$T^{+}(x_{i})$ by $T^{+}(x_{i})\cap Y_{\mathcal{B}}$, but as the difference is
a set of measure zero, it has no effect on the $L^{2}$-space. Secondly, notice
that $\mathcal{H}_{\mathcal{B}}$ comes with a natural
$\mathbb{Z}_{2}$-grading. The associated grading operator is the identity on
$\mathcal{H}^{max}_{\mathcal{B}}$ and minus the identity on
$\mathcal{H}^{min}_{\mathcal{B}}$. Finally, it is a consequence of 7.11 that

$\displaystyle\Delta_{s}:\bigcup_{1\leq i\leq
I_{\mathcal{B}}^{+}}T^{+}(x_{i})$ $\displaystyle\rightarrow$
$\displaystyle\bigcup_{I_{\mathcal{B}}^{+}<j\leq
I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}T^{+}(x_{j})$
$\displaystyle\Delta_{s}:\bigcup_{I_{\mathcal{B}}^{+}<j\leq
I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}T^{+}(x_{j})$
$\displaystyle\rightarrow$ $\displaystyle\bigcup_{1\leq i\leq
I_{\mathcal{B}}^{+}}T^{+}(x_{i})$

are measure preserving bijections and hence induce unitary operators on the
associated $L^{2}$-spaces. In addition, $\Delta_{s}\circ\Delta_{s}$ is the
identity so $F_{\mathcal{B}}$ is odd, $F_{\mathcal{B}}^{2}=1$ and
$F_{\mathcal{B}}=F_{\mathcal{B}}^{*}$.

We need to set out some notation. If $p$ is any element of $E_{m,n}$, we
define

$\displaystyle\xi_{p}^{max}$ $\displaystyle=$
$\displaystyle\nu_{r}(s(p))^{-1/2}\chi_{X_{s(p)}^{-}px_{v}^{s-max}},$
$\displaystyle\xi_{p}^{min}$ $\displaystyle=$
$\displaystyle\nu_{r}(s(p))^{-1/2}\chi_{X_{s(p)}^{-}px_{v}^{s-min}}.$

Each is a unit vector in $\mathcal{H}^{max}_{\mathcal{B}}$ and
$\mathcal{H}^{min}_{\mathcal{B}}$, respectively. Observe that if $e$ is the
$s$-maximal ($s$-minimal) edge with $s(e)=r(p)$, then
$\xi_{p}^{max}=\xi_{pe}^{max}$ ($\xi_{p}^{min}=\xi_{pe}^{min}$, respectively).
It is an easy exercise to check that the linear span of all such vectors is
dense in $\mathcal{H}_{\mathcal{B}}$.

If $\xi,\eta$ are any vectors in a Hilbert space $\mathcal{H}$, we define
$\xi\otimes\eta^{*}$ to be the rank one operator defined by
$(\xi\otimes\eta^{*})\zeta=\langle\zeta,\eta\rangle\xi$, for $\zeta$ in
$\mathcal{H}$. If $T$ is any other operator, we have
$T(\xi\otimes\eta^{*})=(T\xi)\otimes\eta^{*}$ and
$(\xi\otimes\eta^{*})T=(\xi)\otimes(T^{*}\eta)^{*}$.

It is worth noting that it is a straightforward computation from the
definitions that, for any $m<n\leq n^{\prime}$, $p,q$ in $E_{m,n}$
$q^{\prime}$ in $E_{m,n^{\prime}}$, if we let $\xi^{max}_{q^{\prime}}$ be as
above and $a_{p,q}$ be as in 8.2, then
$\pi_{\mathcal{B}}(a_{p,q})\xi_{q^{\prime}}=0$ if $(q^{\prime})_{(m,n]}\neq q$
and $\pi_{\mathcal{B}}(a_{p,q})\xi^{max}_{q^{\prime}}=p^{\prime}$ if
$(q^{\prime})_{(m,n]}=q$, where $p^{\prime}=p(q^{\prime})_{(n,n^{\prime}]}$.
An analogous statement holds for $\xi^{min}_{q^{\prime}}$. In particular, the
representation respects the grading on $\mathcal{H}_{\mathcal{B}}$. In
addition, it will be useful to have the following which is slightly less
routine.

######  Lemma 9.2.

Let $m<n$, $p,q$ in $E_{m,n}$ with $r(p)=r(q)=v$ be in $V_{n}$ and
$\tilde{a}_{p,q}:X_{v}^{+}\rightarrow\mathbb{C}a_{p,q}$ be continuous. For any
$n^{\prime}>n$ and $q^{\prime}$ in $E_{m,n^{\prime}}$, we have

$\pi_{\mathcal{B}}(\tilde{a}_{p,q})\xi_{q^{\prime}}^{max}=\tilde{a}_{p,q}(q^{\prime}x_{r(q^{\prime})}^{s-max})\xi_{p(q^{\prime})_{(n,n^{\prime}]}}^{max}$

if $(q^{\prime})_{(m,n]}=q$ and is zero otherwise, while

$\pi_{\mathcal{B}}(\tilde{a}_{p,q})\xi_{q^{\prime}}^{min}=\tilde{a}_{p,q}(q^{\prime}x_{r(q^{\prime})}^{s-min})\xi_{p(q^{\prime})_{(n,n^{\prime}]}}^{min}$

if $(q^{\prime})_{(m,n]}=q$ and is zero otherwise.

###### Proof.

We prove the first statement only. Let us consider all paths
$p^{\prime\prime}$ in $E_{n,n^{\prime\prime}}$ with
$s(p^{\prime\prime})=r(p)$. We regard
$a_{pp^{\prime\prime},qp^{\prime\prime}}=a_{p,q}\chi_{p^{\prime\prime}X_{r(p^{\prime\prime})}^{+}}$
as a continuous function from $X_{v}^{+}$ to $\mathbb{C}a_{p,q}$. Without loss
of generality, we may assume that $n^{\prime\prime}>n^{\prime}$. Each
continuous function may be approximated by sums of such functions and so it
suffices for us to prove the result for these functions. We have
$\pi(a_{pp^{\prime\prime},qp^{\prime\prime}})\xi_{q^{\prime}}^{max}$ is zero
unless $q=(q^{\prime})_{(m,n]},p^{\prime\prime}=(q^{\prime})_{(n,n^{\prime}]}$
and $p^{\prime\prime}_{(n^{\prime},n^{\prime\prime}]}$ is s-maximal. In this
case the result is $\xi_{p(q^{\prime})_{(n,n^{\prime}]}}^{max}$. In either
case, this agrees with
$\chi_{p^{\prime\prime}X_{r(p^{\prime\prime})}^{+}}(x_{r(q^{\prime})}^{s-max})\xi_{p(q^{\prime})_{(n,n^{\prime}]}}^{max}$.
∎

######  Proposition 9.3.

Let $m<n$ and assume that $p,q$ are in $E_{m,n}^{Y}$ with $r(p)=r(q)=v$.

  1. (1)

We have

$\displaystyle[\pi(a_{p,q}),F_{\mathcal{B}}]$ $\displaystyle=$
$\displaystyle-(F_{\mathcal{B}}\xi_{p}^{max})\otimes(\xi^{max}_{q})^{*}+\xi_{p}^{min}\otimes(F_{\mathcal{B}}\xi^{min}_{q})^{*}$
$\displaystyle-(F_{\mathcal{B}}\xi_{p}^{min})\otimes(\xi^{min}_{q})^{*}+\xi_{p}^{max}\otimes(F_{\mathcal{B}}\xi^{max}_{q})^{*}$
$\displaystyle=$
$\displaystyle[\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*},F_{\mathcal{B}}]+[\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*},F_{\mathcal{B}}]$

  2. (2)

Consider the function
$\tilde{a}_{p,q}(x)=\nu_{r}(v)^{-1}\varphi^{v}_{s}(x)a_{p,q}$, for $x$ in
$X_{v}^{+}$, which is in $AC^{Y+}_{m,n}$. We have

$\displaystyle[\pi_{\mathcal{B}}(\tilde{a}_{p,q}),F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle-(F_{\mathcal{B}}\xi_{p}^{max})\otimes(\xi^{max}_{q})^{*}+\xi_{p}^{max}\otimes(F_{\mathcal{B}}\xi^{max}_{q})^{*}$
$\displaystyle=$
$\displaystyle[\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*},F_{\mathcal{B}}].$

  3. (3)

If $f$ is any continuous $\mathbb{C}$-valued function on $[0,\nu_{r}(v)]$ and
$\tilde{a}_{p,q}(x)=f(\varphi^{v}_{s}(x))a_{p,q}$ then

$\displaystyle[\pi_{\mathcal{B}}(\tilde{a}_{p,q}),F_{\mathcal{B}}]$
$\displaystyle=$ $\displaystyle
f(0)[\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*},F_{\mathcal{B}}]$
$\displaystyle+f(\nu_{r}(v))[\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*},F_{\mathcal{B}}]$

###### Proof.

We consider the first part. Let $\mathcal{H}_{m}$ denote the closed linear
span of all vectors $\xi^{max}_{p},\xi^{max}_{p}$, where $p$ is in
$E_{m,n^{\prime}}$ and $n^{\prime}>m$. It is clear that this space is
invariant under $F_{\mathcal{B}}$ and a direct computation shows that
$\pi(a_{p,q})|_{\mathcal{H}_{m}}=0$. It follows that
$[\pi(a_{p,q}),F_{\mathcal{B}}]|_{\mathcal{H}_{m}}=0$.

Next, let us consider $n^{\prime}>n$ and $q^{\prime}$ in $E_{m,n^{\prime}}$
such that $(q^{\prime})_{(n,n^{\prime}]}$ is not s-maximal. It follows that
$(S_{s}(q^{\prime}))_{(m,n]}=(q^{\prime})_{(m,n]}$ and in consequence
$\pi_{\mathcal{B}}(a_{p,q})F_{\mathcal{B}}\xi_{q^{\prime}}^{max}=\pi_{\mathcal{B}}(a_{p,q})\xi^{min}_{S_{s}(q^{\prime})}$.
If $q\neq(S_{s}(q^{\prime}))_{(m,n]}=(q^{\prime})_{(m,n]}$, this is zero. If
$q=(S_{s}(q^{\prime}))_{(m,n]}=(q^{\prime})_{(m,n]}$, this equals
$\xi^{min}_{p^{\prime}}$ where
$p^{\prime}=p(S_{s}(q^{\prime})_{(n,n^{\prime}]})$. On the other hand,
$F_{\mathcal{B}}\pi_{\mathcal{B}}(a_{p,q})\xi_{q^{\prime}}^{max}$ is also zero
if $q\neq(q^{\prime})_{(m,n]}$, and if $q=(q^{\prime})_{(m,n]}$, it equals
$F_{\mathcal{B}}\xi^{max}_{p^{\prime\prime}}=\xi^{min}_{S_{s}(p^{\prime\prime})}$,
where $p^{\prime\prime}=p(q^{\prime})_{(n,n^{\prime}]}$. As
$(q^{\prime})_{(n,n^{\prime}]}$ is not s-maximal, we have
$S_{s}(p^{\prime\prime})=pS_{s}((q^{\prime})_{(n,n^{\prime}]})=p^{\prime}$. We
conclude that
$[\pi_{\mathcal{B}}(a_{p,q}),F_{\mathcal{B}}]\xi^{max}_{q^{\prime}}=0$. A
similar argument for $\xi^{max}_{q^{\prime}}$ shows the same conclusion.

As we noted above if $q^{\prime}$ is in $E_{m,n^{\prime}},n^{\prime}>n$ and
$(q^{\prime})_{(n,n^{\prime}]}$ is s-maximal, then
$\xi^{max}_{q^{\prime}}=\xi^{max}_{(q^{\prime})_{(n,n^{\prime}]}}$ and so it
remains to consider the case $q^{\prime}$ is in $E_{m,n}$. We need to consider
$[\pi_{\mathcal{B}}(a_{p,q}),F_{\mathcal{B}}]$ on the two types of vectors,
$\xi^{max}_{q^{\prime}}$ and $\xi^{min}_{q^{\prime}}$. Using the fact that
$p,q$ are in $E^{Y}_{m,n}$, we may summarize the only situations where the
result is non-zero as follows:

$\begin{array}[]{rlrr}\pi_{\mathcal{B}}(a_{p,q})F_{\mathcal{B}}\xi_{q^{\prime}}^{max}&=&\xi^{min}_{p},&S_{s}(q^{\prime})=q\\\
\pi_{\mathcal{B}}(a_{p,q})F_{\mathcal{B}}\xi_{q^{\prime}}^{min}&=&\xi^{max}_{p},&P_{s}(q^{\prime})=q\\\
F_{\mathcal{B}}\pi_{\mathcal{B}}(a_{p,q})\xi_{q^{\prime}}^{max}&=&\xi^{min}_{S_{s}(p)},&q^{\prime}=q\\\
F_{\mathcal{B}}\pi_{\mathcal{B}}(a_{p,q})\xi_{q^{\prime}}^{min}&=&\xi^{max}_{P_{s}(p)},&q^{\prime}=q.\end{array}$

Hence, we have

$\displaystyle[\pi_{\mathcal{B}}(a_{p,q}),F_{\mathcal{B}}]$ $\displaystyle=$
$\displaystyle-(\xi_{S_{s}(p)}^{min})\otimes(\xi^{max}_{q})^{*}+\xi_{p}^{min}\otimes(\xi^{max}_{P_{s}(q)})^{*}$
$\displaystyle-(\xi_{P_{s}(p)}^{max})\otimes(\xi^{min}_{q})^{*}+\xi_{p}^{max}\otimes(\xi^{min}_{S_{s}(q)})^{*}$

and recalling that $F_{\mathcal{B}}\xi_{p}^{min}=\xi_{P_{s}(p)}^{max}$ and
$F_{\mathcal{B}}\xi_{p}^{max}=\xi_{S_{s}(p)}^{min}$ completes the proof of the
first equality. The second follows immediately.

The proof for the second part is almost the same. In view of Lemma 9.2, the
operators $\pi(a_{p,q})$ and $\pi_{\mathcal{B}}(\tilde{a}_{p,q})$ are equal
except that

$\pi_{\mathcal{B}}(\tilde{a}_{p,q})\xi_{q}^{max}=\xi_{q}^{max},\pi_{\mathcal{B}}(\tilde{a}_{p,q})\xi_{q}^{min}=0.$

We omit the remaining details.

For the last part, the property is clearly linear in the function $f$ and we
know it is satisfied by constant functions from part 1 and $f(t)=t$ by part 2.
We then show it holds for $f(t)=t^{k},k\geq 1$, by induction on $k$ by noting
that

$\displaystyle[\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k+1}a_{p,q}),F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle[\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q})\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q}),F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle[\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q}),F_{\mathcal{B}}]\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q})$
$\displaystyle+\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q})[\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q}),F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle\nu_{r}(v)^{k}[\xi^{max}_{p}\otimes\xi^{max}_{q},F_{\mathcal{B}}]\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q})$
$\displaystyle+\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q})\nu_{r}(v)[\xi^{max}_{q}\otimes\xi^{max}_{q},F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle\nu_{r}(v)^{k}(\xi^{max}_{p}\otimes\xi^{min}_{S_{s}(q)})\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q})$
$\displaystyle-\nu_{r}(v)^{k}(\xi^{min}_{S_{s}(p)}\otimes\xi^{max}_{q})\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)a_{q,q})$
$\displaystyle+\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q})\nu_{r}(v)(\xi^{max}_{q}\otimes\xi^{min}_{S_{s}(q)})$
$\displaystyle-\pi_{\mathcal{B}}(\varphi^{v}_{s}(x)^{k}a_{p,q})\nu_{r}(v)(\xi^{min}_{S_{s}(q)}\otimes\xi^{max}_{q})$
$\displaystyle=$ $\displaystyle
0-\nu_{r}(v)^{k+1}(\xi^{min}_{S_{s}(p)}\otimes\xi^{max}_{q})$
$\displaystyle+\nu_{r}(v)^{k+1}(\xi^{max}_{p}\otimes\xi^{min}_{S_{s}(q)})-0$
$\displaystyle=$
$\displaystyle\nu_{r}(v)^{k+1}[\xi^{max}_{p}\otimes\xi^{max}_{q},F_{\mathcal{B}}].$

It follows that the result holds for all polynomial functions $f$, and hence
for all continuous functions by continuity. ∎

######  Corollary 9.4.

The triple $(\mathcal{H}_{\mathcal{B}},\pi_{\mathcal{B}},F_{\mathcal{B}})$ is
an even Fredholm module for $A^{Y+}_{\mathcal{B}}$.

######  Lemma 9.5.

Let $m<n$ and for each $p,q$ in $E^{Y}_{m,n}$ with $r(p)=r(q)$, let
$\alpha_{p,q}$ be a complex number. We have

$\displaystyle\left\|\sum_{(p,q)\notin\alpha_{1}(G_{m,n})}\alpha_{p,q}\xi_{p}^{max}\otimes\xi_{q}^{max*}+\sum_{(p,q)\in
G_{m,n}}\frac{\alpha_{p_{1},q_{1}}-\alpha_{p_{2},q_{2}}}{2}\xi_{p}^{max}\otimes\xi_{q}^{max*}\right\|$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\left\|\left[\sum_{r(p)=r(q)}\alpha_{p,q}\xi_{p}^{max}\otimes\xi_{q}^{max*},F_{\mathcal{B}}\right]\right\|$

and

$\displaystyle\left\|\sum_{(p,q)\notin\alpha_{2}(G_{m,n})}\alpha_{p,q}\xi_{p}^{min}\otimes\xi_{q}^{min*}+\sum_{(p,q)\in
G_{m,n}}\frac{\alpha_{p_{2},q_{2}}-\alpha_{p_{1},q_{1}}}{2}\xi_{p}^{min}\otimes\xi_{q}^{min*}\right\|$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\left\|\left[\sum_{r(p)=r(q)}\alpha_{p,q}\xi_{p}^{min}\otimes\xi_{q}^{min*},F_{\mathcal{B}}\right]\right\|.$

###### Proof.

We will prove the first statement only. Let
$\mathcal{F}^{max}=span\\{\xi_{p}^{max}\otimes(\xi^{max}_{q})^{*}\mid p,q,\in
E_{m,n}\\}$ which is a finite dimensional $C^{*}$-algebra.

For each $v$ in $V_{n}$, let
$P_{v}=\sum\xi_{p}^{max}\otimes(\xi^{max}_{p})^{*}$, where the sum is taken
over all $p$ in $E^{Y}_{m,n}$ with $r(p)=v$. Then the map
$\varepsilon:\mathcal{F}^{max}\rightarrow\mathcal{F}^{max}$ defined by
$\varepsilon(a)=\sum_{v\in V_{n}}P_{v}aP_{v}$ is a conditional expectation
from $\mathcal{F}^{max}$ onto $span\\{\xi_{p}^{max}\otimes\xi_{q}^{max*}\mid
r(p)=r(q)\\}$. In particular, $\varepsilon$ is a contraction. Furthermore, for
each $v=(v_{1},v_{2})$ in $r(E^{s}_{m,n})$, we let
$Q_{v}=\sum\xi_{p_{1}}^{max}\otimes(\xi^{max*}_{p_{1}})^{*}$, where the sum is
over all $(p_{1},p_{2})$ in $E_{m,n}^{s}$ with $r(p_{1},p_{2})=v$. Then the
map $\varepsilon^{\prime}:\mathcal{F}^{max}\rightarrow\mathcal{F}^{max}$
defined by $\varepsilon^{\prime}(a)=\sum_{v\in r(E^{s}_{m,n})}Q_{v}aQ_{v}$ is
a conditional expectation from $\mathcal{F}^{max}$ onto
$span\\{\xi_{p_{1}}^{max}\otimes\xi_{q_{1}}^{max*}\mid(p,q)\in G_{m,n}\\}$. In
particular, $\varepsilon^{\prime}$ is a contraction.

A direct computation using Proposition 9.3 shows that

$\displaystyle a^{\prime}$ $\displaystyle=$ $\displaystyle
F_{\mathcal{B}}[\sum_{r(p)=r(q)}\alpha_{p,q}\xi_{p}^{max}\otimes\xi_{q}^{max*},F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle\sum\alpha_{p,q}\left[-\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*}+\xi_{P^{s}(p)}^{max}\otimes(\xi_{P^{s}(q)}^{max})^{*}\right.$
$\displaystyle+\left.\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*}-\xi_{S_{s}(p)}^{min}\otimes(\xi_{S_{s}(q)}^{min})^{*}\right]$

where the sum is over $p,q$ in $E^{Y}_{m,n}$ with $r(p)=r(q)$. Next, we
compute $\varepsilon^{max}(a^{\prime})$. The effect on the last two terms in
the sum is to make them zero, as the vectors do not lie in
$\mathcal{F}^{max}$. The first term is unchanged and the second becomes zero
if $r(P^{s}(p))\neq r(P^{s}(q))$ and is unchanged if
$(p,q)=\alpha_{2}((P^{s}(p)),P^{s}(q)),(p,q))$. Hence, by simply re-indexing
the terms, we have

$\varepsilon(a^{\prime})=\sum_{(p,q)\notin\alpha_{1}(G_{m,n})}\alpha_{p,q}\xi_{p}^{max}\otimes\xi_{q}^{max*}+\sum_{(p,q)\in
G_{m,n}}(\alpha_{p_{1},q_{1}}-\alpha_{p_{2},q_{2}})\xi_{p}^{max}\otimes\xi_{q}^{max*}$

Applying $\varepsilon^{\prime}$ simply removes the first term, so we can write

$\displaystyle\sum_{(p,q)\notin\alpha_{1}(G_{m,n})}\alpha_{p,q}\xi_{p}^{max}\otimes\xi_{q}^{max*}+\sum_{(p,q)\in
G_{m,n}}\frac{\alpha_{p_{1},q_{1}}-\alpha_{p_{2},q_{2}}}{2}\xi_{p}^{max}\otimes\xi_{q}^{max*}$
$\displaystyle=$
$\displaystyle\varepsilon(a^{\prime})-2^{-1}\varepsilon^{\prime}(\varepsilon(a^{\prime})).$

The conclusion follows from the facts that $\varepsilon,\varepsilon^{\prime}$
are contractions. ∎

######  Theorem 9.6.

An element $a$ in $A^{Y+}_{\mathcal{B}}$ is in $B_{\mathcal{B}}^{+}$ if and
only if $[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]=0$.

###### Proof.

Let us begin by proving that if $a$ is in $B_{\mathcal{B}}^{+}$, then
$[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]=0$. To do so, it suffices to assume
that $a=\sum_{p,q}\tilde{a}_{p,q}$ is in $B_{m,n}^{+}$, for some $m<n$: the
general case then follows from part 3 of Theorem 8.9 and continuity.

From the first part of Proposition 8.8 that, for every $p,q$,
$\tilde{a}_{p,q}(x)=\tilde{b}_{p,q}(\varphi^{v}_{r}(x))$, where
$\tilde{b}_{p,q}:[0,\nu_{r}(v)]\rightarrow\mathbb{C}$ is continuous. It
follows that part 3 of Proposition 9.3 applies. Then the expression for
$[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]$ involves a sum of terms and each has
a coefficient of $\tilde{a}_{p,q}(x_{r(p)}^{s-max})$ or
$\tilde{a}_{p,q}(x_{r(p)}^{s-min})$.

We observe from Lemma 8.8 that $\tilde{a}_{p,q}(x_{r(p)}^{s-max})$ is non-zero
only if there exists $(p,p^{\prime}),(q,q^{\prime})$ in $E^{s}_{m,n}$ and, in
this case, $p^{\prime}=S_{s}(p),q^{\prime}=S_{s}(q)$ are unique. Similarly,
$\tilde{a}_{p,q}(x_{r(p)}^{s-min})$ is non-zero only if there exists
$(p^{\prime},p),(q^{\prime},q)$ in $E^{s}_{m,n}$ and, in this case,
$p^{\prime}=P^{s}(p),q^{\prime}=P^{s}(q)$ are unique. We conclude from this
that

$\displaystyle[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]$ $\displaystyle=$
$\displaystyle\sum_{p,q\in
E^{s}_{m,n}}\tilde{a}_{p_{1},q_{1}}(x_{r(p_{1})}^{s-max})[\xi_{p_{1}}^{max}\otimes(\xi_{q_{1}}^{max})^{*},F_{\mathcal{B}}]$
$\displaystyle+\tilde{a}_{p_{2},q_{2}}(x_{r(p_{2})}^{s-min})[\xi_{p_{1}}^{min}\otimes(\xi_{q_{2}}^{min})^{*},F_{\mathcal{B}}]$
$\displaystyle=$
$\displaystyle\tilde{a}_{p_{1},q_{1}}(x_{r(p_{1})}^{s-max})\xi_{p_{1}}^{max}\otimes(\xi_{q_{2}}^{min})^{*}$
$\displaystyle-\tilde{a}_{p_{1},q_{1}}(x_{r(p_{1})}^{s-max})\xi_{p_{2}}^{min}\otimes(\xi_{q_{1}}^{max})^{*}$
$\displaystyle+\tilde{a}_{p_{2},q_{2}}(x_{r(p_{2})}^{s-min})\xi_{p_{2}}^{min}\otimes(\xi_{q_{1}}^{max})^{*}$
$\displaystyle\tilde{a}_{p_{2},q_{2}}(x_{r(p_{2})}^{s-min})\xi_{p_{1}}^{max}\otimes(\xi_{q_{2}}^{min})^{*}$
$\displaystyle=$ $\displaystyle 0$

by the last condition of Proposition 8.8.

For the converse direction, from the facts that the union of the
$A^{Y+}_{m,n}$ are dense in $A^{Y+}_{\mathcal{B}}$ and the function sending
$a$ in $A^{Y+}_{\mathcal{B}}$ to $\|[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]\|$
is continuous, it suffices for us to prove that, for any $m<n$ and $a$ in
$A^{Y+}_{m,n}$, there is $b$ in $AC^{Y+}_{m,n}$ with $a-b$ in $B_{m,n}^{+}$
and $\|b\|\leq 2\|[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]\|$.

Let $a=\sum_{p,q}\alpha_{p,q}a_{p,q}$ be in $A_{m,n}^{Y+}$, where the sum is
over $p,q$ in $E_{m,n}^{Y}$ with $r(p)=r(q)$. For each $v$ in $V_{n}$, define
$c:X_{v}^{+}\rightarrow A_{m,n,v}^{+}$ by

$\displaystyle b(x)$ $\displaystyle=$
$\displaystyle\nu_{r}(v)^{-1}\varphi^{v}_{r}(x)\left(\sum_{r(p)=r(q),(p,q)\notin\alpha_{1}(G_{m,n})}\alpha_{p,q}a_{p,q}\right.$
$\displaystyle+\left.\sum_{(p,q)\in
G_{m,n}}\frac{\alpha_{p_{1},q_{1}}-\alpha_{p_{2},q_{2}}}{2}a_{p_{1},q_{1}}\right)$
$\displaystyle+(1-\nu_{r}(v)^{-1}\varphi^{v}_{r}(x))\left(\sum_{r(p)=r(q),(p,q)\notin\alpha_{2}(G_{m,n})}\alpha_{p,q}a_{p,q}\right.$
$\displaystyle+\left.\sum_{(p,q)\in
G_{m,n}}\frac{\alpha_{p_{2},q_{2}}-\alpha_{p_{1},q_{1}}}{2}a_{p_{2},q_{2}}\right)$

for $x$ in $X_{v}^{+}$. We regard $c$ as an element of $AC^{Y+}_{m,n}$.

It is a simple computation, using the results of Propositions 9.3 and 8.8 to
verify that $a-b$ is in $B_{m,n}$. It remains for us to prove that $\|b\|\leq
2\|[\pi_{\mathcal{B}}(a),F_{\mathcal{B}}]\|$.

The map, which we denote by $\eta^{max}$, sending $a_{p,q}$ to
$\xi_{p}^{max}\otimes(\xi^{max}_{q})^{*}$, for $p,q$ in $E^{Y}_{m,n}$ with
$r(p)=r(q)$, extends linearly to an injective $*$-homomorphism from
$A^{Y+}_{m,n}$ to $\mathcal{F}^{max}$ which is necessarily isometric. The
desired inequality follows from this and two applications of Lemma 9.5. ∎

##  10\. $K$-theory

The purpose of this section is to compute the $K$-theory of the
$C^{*}$-algebras considered in the section 8. It is probably more accurate to
say that we shall investigate the relations between the $K$-theory of the
$C^{*}$-algebras.

We will continue to assume the standing hypotheses of 6.2.

Given a bi-infinite Bratteli diagram $\mathcal{B}$, the $K$-theory of the AF-
algebra $A_{\mathcal{B}}^{+}$ is readily computable from the data given and
the results of Proposition 8.3. It is worth noting at this point that it does
not depend on the order structure, nor the half of the diagram indexed by the
negative integers.

######  Theorem 10.1.

Let $\mathcal{B}$ be a bi-infinite Bratteli diagram. For each integer $n$, we
consider $E_{n}$ to be the $\\#V_{n}\times\\#V_{n-1}$ positive integer matrix
which describes the edge set $E_{n}$. We have

$K_{0}(A_{\mathcal{B}}^{+})\cong\lim_{n\rightarrow+\infty}\mathbb{Z}^{\\#V_{0}}\stackrel{{\scriptstyle
E_{1}}}{{\rightarrow}}\mathbb{Z}^{\\#V_{1}}\stackrel{{\scriptstyle
E_{2}}}{{\rightarrow}}\cdots$

and $K_{1}(A_{\mathcal{B}}^{+})=0$.

######  Theorem 10.2.

Let $\mathcal{B}$ be an ordered bi-infinite Bratteli diagram. Then the
inclusion $A^{Y+}_{\mathcal{B}}\subseteq A_{\mathcal{B}}^{+}$ induces an order
isomorphism $K_{0}(A^{Y+}_{\mathcal{B}})\cong K_{0}(A_{\mathcal{B}}^{+})$. and
$K_{1}(A^{Y+}_{\mathcal{B}})=0$.

We now turn to the $C^{*}$-algebra $B_{\mathcal{B}}$.

######  Proposition 10.3.

Let $m<n$, $v$ be any vertex in $V_{n}$, $p$ be in $E_{m,n}^{Y}$ with $r(p)=v$
and $f_{v}:[0,\nu_{r}(v)]\rightarrow[0,1]$ be any continuous function with
$f_{v}(0)=0,f(\nu_{r}(v))=1$. Then $K_{1}(B_{\mathcal{B}}^{+})\cong\mathbb{Z}$
and is generated by the unitary $u=exp(2\pi
if_{v}\circ\varphi_{r}^{v}(x))a_{p,p}+(1-a_{p,p})$ considered as an element of
$\tilde{B^{+}}_{m,n}\subseteq\tilde{B^{+}}_{\mathcal{B}}$.

###### Proof.

We use the fact that $B_{\mathcal{B}}^{+}$ is the closure of the union of the
$B_{m,n}^{+},m<n$, so

$K_{1}(B_{\mathcal{B}}^{+})=\lim_{n\rightarrow\infty}K_{1}(B_{-n,n}^{+}).$

We will first compute $K_{1}(B_{-n,n}^{+})$ and then the inductive limit.

We use with the short exact sequence found in Corollary 8.10

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(G_{m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

For simplicity, we denote $A_{m,n,v}^{+}\otimes C_{0}(0,\nu_{s}(v))$ by
$\mathcal{I}_{v}$. We have the associated six-term exact sequence for
$K$-groups

$\textstyle{K_{0}(\oplus_{v\in
V_{n}}\mathcal{I}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(B_{m,n}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(C^{*}(G_{m,n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(C^{*}(G_{m,n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(B_{m,n}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(\oplus_{v\in
V_{n}}\mathcal{I}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Let us start with
$K_{*}(\oplus_{v}\mathcal{I}_{v})\cong\bigoplus_{v}K_{*}(\mathcal{I}_{v})$. As
$A_{m,n,v}^{+}$ is a full matrix algebra, we have $K_{0}(\mathcal{I}_{v})\cong
K_{0}(C_{0}(0,\nu_{v}(r)))\cong K_{1}(\mathbb{C})=0$ while
$K_{1}(\mathcal{I}_{v})\cong K_{1}(C_{0}(0,\nu_{v}(r)))\cong
K_{0}(\mathbb{C})\cong\mathbb{Z}$. Moreover, if $p,f_{v}$ are as above, then
$u_{v}=exp(2\pi if\circ\varphi_{r}^{v}(x))a_{p,p}+(1-a_{p,p})$ is a generator
of this group.

We now turn to $K_{*}(C^{*}(G_{-n,n}))$. The groupoid $G_{-n,n}$ is finite and
its $C^{*}$-algebra is finite-dimensional. Hence its $K_{1}$-group is trivial.
On the other hand, it is a direct sum a full matrix algebras, indexed by the
elements of $r(p),p\in E^{s}_{-n,n}$. It follows that
$K_{*}(C^{*}(G_{-n,n}))\cong\bigoplus_{r(E^{s}_{-n,n})}\mathbb{Z}$, with
generators $[\chi_{(p,p)}]_{0}$, where $p$ is chosen to be any path in
$E^{s}_{-n,n}$, as $r(p)$ takes all possible values.

Our six-term exact sequence now looks like

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(B_{-n,n}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{r(E^{s}_{-n,n})}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{exp}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(B_{-n,n}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

It is a fairly standard argument to check that the exponential map takes
$[\chi_{(p,p)}]_{0}$ in $K_{0}(C^{*}(G_{-n,n}))$, where $p$ is any path in
$E^{s}_{-n,n}$, to $[u_{p_{1}}]_{1}-[u_{p_{2}}]$ in
$\bigoplus_{v}K_{1}(\mathcal{I}_{v})$.

From this we can see that the exponential map is not surjective; indeed for
any fixed $v$, the elements $m[u_{v}]_{1}$ are all distinct in
$K_{1}(B_{-n,n}^{+})$.

To compute the inductive limit, it suffices to show that, for $v$ in $V_{n}$
and $v^{\prime}$ in $V_{n^{\prime}}$ with $n^{\prime}>n$, we have
$[u_{v}]_{1}=[u_{v^{\prime}}]_{1}$, as elements of
$K_{1}(B_{m,n^{\prime}}^{+})$, provided that there is at least one path
$p^{\prime}$ from $v$ to $v^{\prime}$. Let $p$ be any element of $E_{m,n}$
with $r(p)=v$ and let $f_{v^{\prime}}$ be any function as above. Then define
$f_{v}$ as follows

$f_{v}(x)=\left\\{\begin{array}[]{cl}0&x_{(n,n^{\prime}]}<_{s}p^{\prime}\\\
f_{v^{\prime}}(x_{(n^{\prime},\infty)})&x_{(n,n^{\prime}]}=p^{\prime}\\\
1&x_{(n,n^{\prime}]}<_{s}p^{\prime}\end{array}\right.$

It is easy to see that $f_{v}$ satisfies the desired conditions and that, with
these choices, $u_{v}=u_{v^{\prime}}$. ∎

To describe the $K$-zero group, we need to establish some notation.

For any finite set $A$, let $\mathbb{Z}A$ denote the free abelian group on
$A$. Recalling the definition of
$I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$ from Definition 7.5, we
define

$\displaystyle\theta_{1}:\mathbb{Z}(I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+})$
$\displaystyle\rightarrow$
$\displaystyle\mathbb{Z}\\{x_{1},\ldots,x_{I_{\mathcal{B}}^{+}}\\},$
$\displaystyle\theta_{2}:\mathbb{Z}(I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+})$
$\displaystyle\rightarrow$
$\displaystyle\mathbb{Z}\\{x_{I_{\mathcal{B}}^{+}+1},\ldots,x_{J_{\mathcal{B}}^{+}}\\},$
$\displaystyle\theta:\mathbb{Z}(I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+})$
$\displaystyle\rightarrow$
$\displaystyle\mathbb{Z}\\{x_{1},\ldots,x_{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}}^{+}\\}$
$\displaystyle\sigma:\mathbb{Z}\\{x_{1},\ldots,x_{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}\\}$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{Z}$

by $\theta_{1}(x_{i},x_{j})=x_{i},\theta_{2}(x_{i},x_{j})=x_{j}$,
$\theta(x_{i},x_{j})=x_{i}+x_{j}$ and $\sigma(x_{i})=1,1\leq i\leq
I_{\mathcal{B}}^{+},\sigma(x_{j})=-1,I_{\mathcal{B}}^{+}<j\leq
I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}$. Observe that $\sigma\circ\theta=0$,
so $\sigma$ also defines a homomorphism from $coker(\theta)$ to $\mathbb{Z}$.

######  Theorem 10.4.

There is a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ker(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(B_{\mathcal{B}}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(A_{\mathcal{B}}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{coker(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $i:B_{\mathcal{B}}^{+}\rightarrow A_{\mathcal{B}}^{+}$ is the inclusion
map. In particular, $K_{0}(B_{\mathcal{B}}^{+})$ is finite rank and is
finitely generated if and only if $K_{0}(A_{\mathcal{B}}^{+})$ is. If either
$I_{\mathcal{B}}^{+}=1$ or $J_{\mathcal{B}}^{+}=1$, then $i_{*}$ is an
isomorphism.

###### Proof.

We make use of the notion of the relative $K$-theory for $C^{*}$-algebras
along with an excision result of the second author [Put21]. Relative
$K$-theory was introduced by Karoubi [Kar08], but we also refer the reader to
[Put21] or Haslehurst [Has21] for a more extensive treatment. To any
$C^{*}$-algebra, $A$, and $C^{*}$-subalgebra, $A^{\prime}\subseteq A$, there
are relative $K$-groups, $K_{i}(A^{\prime};A),i=0,1$ which fit into a six-term
exact sequence

$\textstyle{K_{0}(A^{\prime};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{1}(A^{\prime};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $i:A^{\prime}\rightarrow A$ denote the inclusion map.

In Theorems 3.2 and 3.4 of [Put21], the situation is described of
$C^{*}$-algebras $A,B,E$ along with a bounded $*$-derivation
$\delta:A+B\rightarrow E$ such that there is a natural isomorphism
$K_{*}(\ker(\delta)\cap A;A)\cong K_{*}(\ker(\delta)\cap B;B)$. Referring back
to notation established in Definition 9.1, we use
$A=\bigoplus_{i=1}^{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}\mathcal{K}(\mathcal{H}_{i})$,
where $\mathcal{K}$ denotes the $C^{*}$-algebra of compact operators,
$B=A^{Y+}_{\mathcal{B}}$ or more accurately,
$B=\pi_{\mathcal{B}}(A^{Y+}_{\mathcal{B}})$. As we noted earlier, the
representation is faithful under our hypotheses, so this amounts to a
notational difference only. We use $E=\mathcal{B}(\mathcal{H}_{\mathcal{B}})$,
the algebra of bounded linear operators on $\mathcal{H}_{\mathcal{B}}$ and
$\delta(x)=i\left[x,F_{\mathcal{B}}\right]$, for any operator $x$. (The use of
$\mathcal{B}$ for the bounded linear operators and for the Bratteli diagram,
is unfortunate, but should not cause any confusion.)

We need to verify the hypotheses of [Put21] hold. The first is that
$AB\subseteq A$ and this follows from the facts that, for all $i$,
$\mathcal{H}_{i}$ is invariant for the representation $\pi_{\mathcal{B}}$ and
that $A$ consists entirely of compact operators on this space.

The hypotheses of Theorem 3.4 of [Put21] involve the choice of a dense
$*$-subalgebra, $\mathcal{A}\subseteq A$. For this, we use the linear span of
all rank one operators of the form $\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*}$
and $\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*}$, where $p,q$ vary over
$E^{Y}_{m,n}$ with $r(p)=r(q)$ and $m<n$ vary over all integers.

We now verify property C1 from Theorem 3.4 of [Put21]: let

$a=\sum\alpha^{max}_{p,q}\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*}+\alpha^{min}_{p,q}\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*},$

where the sum is over $p,q$ in $E^{Y}_{m,n}$ with $r(p)=r(q)$, be in
$\mathcal{A}$. Let

$a^{\prime}=\sum\frac{\alpha^{max}_{p_{1},q_{1}}+\alpha^{min}_{p_{2},q_{2}}}{2}\left(\xi_{p_{1}}^{max}\otimes(\xi_{q_{1}}^{max})^{*}+\xi_{p_{2}}^{min}\otimes(\xi_{q_{2}}^{min})^{*}\right)$

where the sum is over all $((p_{1},q_{1}),(p_{2},q_{2}))$ in $G_{m,n}$. It is
an easy calculation that $\delta(a^{\prime})=i[a^{\prime},F_{\mathcal{B}}]=0$
and $\|a-a^{\prime}\|\leq\frac{3}{2}\|\delta(a)\|$ follows immediately from
the first part of Proposition 9.3 and Lemma 9.5.

Using the dense $*$-subalgebra $\bigcup_{n}A^{Y+}_{-n,n}$ of
$A^{Y+}_{\mathcal{B}}$, and Lemma 4.2 of [Put21], we see that
$\delta(B)\subseteq\delta(A)$.

It remains to see that condition C2 of Theorem 3.4 of [Put21] holds. For that,
we can assume that the $a_{1},\ldots,a_{I}$ all lie in some
$span\\{\xi^{max}_{p}\otimes(\xi^{max}_{q})^{*},\xi^{min}_{p}\otimes(\xi^{min}_{q})^{*}\\}$,
as $p,q$ range over $E^{Y}_{m,n}$, for some $m,n$. We let $e$ be the unit of
this algebra,

$e=\sum_{p\in
E^{Y}_{m,n}}\xi_{p}^{max}\otimes(\xi_{p}^{max})^{*}+\xi_{p}^{min}\otimes(\xi_{p}^{min})^{*}$

and for

$a_{i}=\sum\alpha^{max}_{p,q}\xi_{p}^{max}\otimes(\xi_{q}^{max})^{*}+\alpha^{min}_{p,q}\xi_{p}^{min}\otimes(\xi_{q}^{min})^{*},$

we use $b_{i}$ in $AC^{Y+}_{m,n}$ defined by

$b_{i}(x)=\sum\left(\nu_{r}(r(p))^{-1}\varphi^{r(p)}_{r}(x)\alpha^{max}_{p,q}+(1-\nu_{r}(r(p))^{-1}\varphi^{r(p)}_{r}(x))\alpha^{min}_{p,q}\right)a_{p,q}$

for $x$ in $\bigcup_{v}X_{v}^{+}$. The desired properties follow from
Proposition 9.3; we omit the details. We have verified the conditions of
Theorem 3.4 of [Put21]. In addition, Theorem 9.6 shows that
$B_{\mathcal{B}}^{+}=\ker(\delta)\cap A^{Y+}_{\mathcal{B}}$. We conclude that
conclude that $K_{*}(\ker(\delta)\cap A;A)\cong
K_{*}(B_{\mathcal{B}}^{+};A^{Y+}_{\mathcal{B}})$.

We now turn to the computation of $K_{*}(\ker(\delta)\cap A;A)$. Recall that
$A=\bigoplus_{i=1}^{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}\mathcal{K}(\mathcal{H}_{i})$.
For $(i,j)$ in $I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$, we
define

$\mathcal{H}_{i,j}=\mathcal{H}_{i}\cap
F_{\mathcal{B}}\mathcal{H}_{j}=L^{2}(T^{+}(x_{i})\cap\Delta_{s}(T^{+}(x_{j})))$

and observe that $F_{\mathcal{B}}H_{i,j}=L^{2}(\Delta_{s}(T^{+}(x_{i}))\cap
T^{+}(x_{j})).$ It is a simple matter to check that the map sending
$(k_{i,j})_{(i,j)}$ to
$\sum_{(i,j)}k_{i,j}+F_{\mathcal{B}}k_{i,j}F_{\mathcal{B}}$ is an isomorphism
between
$\oplus_{I_{\mathcal{B}}\star_{\Delta}J_{\mathcal{B}}}\mathcal{K}(H_{i,j})$
and $\ker(\delta)\cap A$.

For any Hilbert space $\mathcal{H}$, there is a canonical isomorphism from
$K_{0}(\mathcal{K}(\mathcal{H}))$ to $\mathbb{Z}$ induced by the trace. In
addition, we have $K_{1}(\mathcal{K}(\mathcal{H}))\cong 0$ (see of ). Hence,
we have $K_{1}(A)\cong K_{1}(\ker(\delta)\cap A)\cong 0$,
$K_{0}(A)\cong\mathbb{Z}\\{x_{1},\ldots,x_{I}\\}\oplus\mathbb{Z}\\{x_{I+1},\ldots,x_{J}\\}$
and $K_{0}(\ker(\delta)\cap
A)\cong\mathbb{Z}I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$.
Moreover, the map induced by the inclusion $\ker(\delta)\cap A\subseteq A$ is
simply $\theta$. In summary, the six-term exact sequence for the relative
groups of the inclusion becomes

$\textstyle{K_{0}(\ker(\delta)\cap
A;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\mathbb{Z}\\{x_{1},\ldots,x_{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(\ker(\delta)\cap
A;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and so $K_{0}(\ker(\delta)\cap A;A)\cong ker(\theta)$ and
$K_{1}(\ker(\delta)\cap A;A)\cong coker(\theta)$. Combining this with the
computation of the relative groups already done above and the results of
Theorem 10.2 and Proposition 10.3 completes the proof.

The remaining statements are straightforward. In particular, it is a simple
matter to check that if $I_{\mathcal{B}}^{+}=1$, then
$I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}=\\{1\\}\times\\{1,\ldots,J_{\mathcal{B}}^{+}\\}$
and $\theta(1,j)=x_{1}+x_{j}$, for $1\leq j\leq J_{\mathcal{B}}^{+}$, which is
clearly injective and has $coker(\theta)\cong\mathbb{Z}$. ∎

A crucial part of K-theory (at least $K_{0}$) for a $C^{*}$-algebra is its
natural order structure. As a simple example, if $\alpha,\beta$ are any two
irrational numbers, then the subgroups of the real numbers
$\mathbb{Z}+\alpha\mathbb{Z}$ and $\mathbb{Z}+\beta\mathbb{Z}$ are isomorphic
as abstract groups, but with the relative orders from the real numbers, they
are not isomorphic in general as ordered groups. One of the difficulties in
operator algebra K-theory is that many computational tools do not respect the
order structure. As an example here, while we may easily check in some
specific situation that the map $i_{*}$ of Theorem 10.4 is an isomorphism, it
does not follow at once that it is an isomorphism of ordered groups. Part of
that is easily dealt with: the fact that it is induced by a $*$-homomorphism
of $B_{\mathcal{B}}^{+}$ in $A_{\mathcal{B}}^{+}$ means that it is a positive
homomorphism in the sense it maps the positive cone in the former into the
positive cone in the latter.

######  Theorem 10.5.

Let $\mathcal{B}$ be an ordered Bratteli diagram satisfying the standard
hypotheses. If the following sequence is exact

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\mathbb{Z}\\{x_{1},\ldots,x_{I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
36.135pt\sigma}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and the equivalence classes of the relation $T^{\sharp}(Y_{\mathcal{B}})$ are
all dense, then

$i_{*}:K_{0}(B_{\mathcal{B}}^{+})\rightarrow K_{0}(A_{\mathcal{B}}^{+})$

is an isomorphism of ordered abelian groups. In particular, if
$I_{\mathcal{B}}^{+}$ or $J_{\mathcal{B}}^{+}$ is equal to one, then the same
conclusion holds.

###### Proof.

We know already from the last theorem and the hypothesis on the exact sequence
that $i_{*}$ is an isomorphism and since it is induced by a $*$-homomorphism
at the level of $C^{*}$-algebras, it maps positive elements to positive
elements. It remains for us to show that every positive element of
$K_{0}(A_{\mathcal{B}}^{+})$ is the image of a positive element of
$K_{0}(B_{\mathcal{B}}^{+})$. In view of Theorems 10.1 and 10.2, it suffices
to consider a projection in $A_{\mathcal{B}}^{+}$ of the form $a_{p,p}$, where
$p$ is in $E_{m,n}^{Y}$, for some $m<n$, and show it is Murray-von Neumann
equivalent to one in $B_{\mathcal{B}}^{+}$.

Consider two points $x,y$ in $X_{r(p)}$ satisfying the following:
$x\leq_{s}y$, $X_{r(p)}^{-}x\subseteq T^{+}(x_{i})$ and
$X_{r(p)}^{-}y\subseteq T^{+}(x_{j})$, for some $1\leq i\leq
I_{\mathcal{B}}^{+}<j\leq I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}$. It follows
that $a_{p,p}\otimes\chi_{[x,y]}$ is in $A^{Y+}_{m,n}\otimes
C(X_{r(p)}^{+})=AC_{m,n}^{Y+}$ and so it determines a class in
$K_{0}(A_{\mathcal{B}}^{+})$.

Observe that as $p$ is not $s$-maximal or $s$-minimal,
$\Delta_{s}(X_{s(p)}^{-}px)_{(m,\infty)}$ is a single path, as is
$\Delta_{s}(X_{s(p)}^{-}py)_{(m,\infty)}$. In particular,
$\Delta_{s}(X_{s(p)}^{-}px)$ is contained in $T^{+}(x_{j^{\prime}})$, for some
$j^{\prime}$, while $\Delta_{s}(X_{s(p)}^{-}py)$ is contained in
$T^{+}(x_{i^{\prime}})$, for some $i^{\prime}$.

We first consider the special case that $j^{\prime}=j$. (The case
$i^{\prime}=i$ can be done in a similar way.) This means we can find $N>n$
such that $\Delta_{s}(X_{s(p)}^{-}px)_{(N,\infty)}=y_{(N,\infty)}$. Let
$\bar{p}=\Delta_{s}(X_{s(p)}^{-}px)_{(m,N]}$.

We define

$\displaystyle w$ $\displaystyle=$
$\displaystyle\sum_{p^{\prime}}a_{p^{\prime},p^{\prime}}+\cos\left(\frac{\pi}{2}\nu_{r}(r(y_{N}))^{-1}\varphi^{r(y_{N})}_{r}(z)\right)a_{y_{(m,N]},y_{(m,N]}}$
$\displaystyle+\sin\left(\frac{\pi}{2}\nu_{r}(r(y_{N}))^{-1}\varphi^{r(y_{N})}_{r}(z)\right)a_{y_{(m,N]},\bar{p}}$

where the sum is over all $p^{\prime}$ in $E_{m,N}$ with
$px\leq_{s}p^{\prime}<_{s}py$ and the variable $z$ lies in $X_{r(y_{N})}^{+}$.
It is a simple matter to check that $w^{*}w=a_{p,p}\otimes\chi_{[x,y]}$ while
$ww^{*}$ lies in $AC_{m,N}^{Y+}$. We conclude that the class of
$a_{p,p}\otimes\chi_{[x,y]}$ lies in the image of $i_{*}$.

We now consider the general case, dropping the hypothesis that $j^{\prime}=i$.
It is clear that $x_{i^{\prime}}-x_{j^{\prime}}$ lies in the kernel of
$\sigma$. It follows that we may find a finite sequence $(i_{l},j_{l}),1\leq
l\leq L$ in $I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$ such that
$j_{1}=j^{\prime},j_{l+1}=i_{l},1\leq l<L$ and $i_{L}=i^{\prime}$. By the
minimality of $T^{+}(x_{i_{1}})\cap\Delta_{s}(T^{+}(x_{j_{1}}))$, we may find
$y_{1}$ in $X^{+}_{r(p)}$ with $x<_{s}y_{1}<_{s}y$ with

$X_{r(p)}^{-}y_{1}\subseteq T^{+}(x_{i_{1}})\cap\Delta_{s}(T^{+}(x_{j_{1}})).$

By application of the special case above, the class of
$a_{p,p}\otimes\chi_{[x,y_{1}]}$ lies in the image of $i_{*}$. Continuing in
this way, we may construct $x<_{s}y_{1}<_{s}y_{2}<_{s}\cdots<_{s}y_{L}<_{s}y$
such that $y_{l}$ is in $T^{+}(x_{i_{l}})\cap\Delta_{s}(T^{+}(x_{j_{l}}))$ and
the class of $a_{p,p}\otimes\chi_{[y_{l},y_{l+1}]}$ and also
$a_{p,p}\otimes\chi_{[y_{L},y]}$ lie in the image of $i_{*}$. We conclude that
the class of

$a_{p,p}\otimes\chi_{[x,y]}=a_{p,p}\otimes\chi_{[x,y_{1}]}+\sum_{l=1}^{L-1}a_{p,p}\otimes\chi_{[y_{l},y_{l+1}]}+a_{p,p}\otimes\chi_{[y_{L},y]}$

also lies in the image of $i_{*}$. Finally, we note that if we choose
$x=x_{r(p)}^{s-min}$ and $y=x_{r(p)}^{s-max}$, then
$a_{p,p}=a_{p,p}\otimes\chi_{[x,y]}$.

We finish by considering the case $I_{\mathcal{B}}^{+}=1$ ( with
$J_{\mathcal{B}}^{+}=1$ being similar). For any $1+I_{\mathcal{B}}^{+}\leq
j\leq I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}$, we know that
$\Delta_{s}(T^{+}(x_{j}))$ must be contained in $T^{+}(x_{1})$, so
$I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}=\\{1\\}\times\\{1+I_{\mathcal{B}}^{+},\ldots,I_{\mathcal{B}}^{+}+J_{\mathcal{B}}^{+}\\}$
and it is a simple matter to verify the given sequence is exact. Next, we also
have
$\Delta_{s}(T^{+}(x_{1}))\cap T^{+}(x_{j})=T^{+}(x_{j})$ which is dense by our
hypotheses on $\mathcal{B}$. It follows that every equivalence class in
$T^{\sharp}(Y_{\mathcal{B}})$ is dense. ∎

We finally turn to the K-theory of the foliation algebra of
$(S_{\mathcal{B}},\mathcal{F}_{\mathcal{B}}^{+})$.

######  Remark 10.6.

As the foliation $\mathcal{F}_{\mathcal{B}}^{+}$ arises from an action of
$\mathbb{R}$ on the space $S_{\mathcal{B}}$, Connes’ analogue of the Thom
isomorphism Theorem (see 10.2.2 of [Bla86]) asserts that

$K_{i}(C^{*}(\mathcal{F}_{\mathcal{B}}^{+}))\cong K^{i+1}(S_{\mathcal{B}}).$

On the other hand, this is not terribly useful at the moment, since we don’t
know the $K$-theory (or cohomology) of the space $S_{\mathcal{B}}$, nor does
it seem particularly likely that it can be computed directly, given our
construction. In any event, Connes’ result does not reveal anything about the
order structure on the $K_{0}$ group of the foliation algebra. Instead, we
will compute its $K$-theory as it relates to our AF-algebra. Having done this,
we can then use Connes’ result to compute the $K$-theory of our surface.

We begin by recalling some notation. We let $\mathcal{I}_{\mathcal{B}}$ be the
collection of connected subsets of the union of
$\pi(T^{+}(x_{i})\cap\Delta_{s}(T^{+}(x_{j})))$ over all $(i,j)$ in
$I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$. We also recall that
each such subset is homeomorphic to $\mathbb{R}$. Now, for each $I$ in
$\mathcal{I}_{\mathcal{B}}$, we define $\iota(I)=(i,j)$, if
$I\subseteq\pi(T^{+}(x_{i})\cap\Delta_{s}(T^{+}(x_{j})))$. It is clearly
surjective. We also let $\iota$ be the map induced from
$\mathbb{Z}\mathcal{I}_{\mathcal{B}}$ to
$\mathbb{Z}(I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+})$.

We are going to construct a sequence of groupoids and $C^{*}$-algebras
interpolating between $\mathcal{F}_{\mathcal{B}}^{+}$ and
$T^{\sharp}(S_{\mathcal{B}})$. Let us begin by selecting
$\mathcal{I}_{0}\subseteq\mathcal{I}_{\mathcal{B}}$ which contains exactly one
interval from each set $\pi(T^{+}(x_{i})\cap\Delta_{s}(T^{+}(x_{j}))$. That
is, $\iota:\mathcal{I}_{0}\rightarrow
I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$ is a bijection. We then
enumerate the remaining intervals of
$\mathcal{I}_{\mathcal{B}}-\mathcal{I}_{0}$ as $I_{1},I_{2},\ldots$. Although
this may be finite, we will ignore that in our notation. Observe that, for
each $l\geq 1$, there is a unique $I^{\prime}_{l}$ in $\mathcal{I}_{0}$ such
that $\iota(I_{l})=\iota(I^{\prime}_{l})$ and the collection
$I_{l}-I^{\prime}_{l}$, $l\geq 1$ is a set of generators for $ker(\iota)$
having no relations.

We define a sequence of groupoids, beginning with
$\mathcal{F}_{0}^{+}=\mathcal{F}_{\mathcal{B}}^{+}$. Then for $l\geq 1$, set
$\mathcal{F}_{l}^{+}$ to be the union of $\mathcal{F}_{l-1}^{+}$ with all sets
$I_{l}\times I$ and $I\times I_{l}$, where $I$ is in $\mathcal{F}_{l-1}^{+}$
and satisfies $\iota(I)=\iota(I_{l})$. That is, on the set $\cup_{j\leq
l}I_{j}$, $\mathcal{F}_{l}^{+}$ agrees with $T^{\sharp}(S_{\mathcal{B}})$,
while on $\cup_{j>l}I_{j}$, it agrees with $\mathcal{F}_{\mathcal{B}}$. We
leave it as a simple exercise to check that $\mathcal{F}_{l}^{+}$ is an open
subgroupoid of $T^{\sharp}(S_{\mathcal{B}})$, $\mathcal{F}_{l}^{+}$ is an open
subgroupoid of $\mathcal{F}_{l+1}^{+}$ and the union over all $l$ is
$T^{\sharp}(S_{\mathcal{B}})$.

We let $j_{l}$ to denote the inclusion of $C^{*}(\mathcal{F}_{l}^{+})$ in
$C^{*}(T^{\sharp}(S_{\mathcal{B}}))$ and $i_{l,k}$ to denote the inclusion of
$C^{*}(\mathcal{F}_{k}^{+})$ in $C^{*}(\mathcal{F}_{l}^{+})$, for $k\leq l$.

######  Theorem 10.7.

Let $l\geq 0$ and let $j$ denotes the inclusion of
$C^{*}(\mathcal{F}_{l}^{+})$ in $C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}}))$, then

$(j_{l})_{*}:K_{1}(C^{*}(\mathcal{F}_{l}^{+}))\rightarrow
K_{1}(C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}})))\cong\mathbb{Z}$

is an isomorphism.

###### Proof.

For $m<n$, we define the groupoid $H_{l,m,n}$ to be all $(p,q)$ in $G_{m,n}$
such that $I(p)=I(q)$ if either equals $I_{j}$, for some $j>l$. Recall the
short exact sequence of Proposition 8.10:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C^{*}(G_{m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

where we have used $q$ to denote the quotient map. As $H_{l,m,n}$ is a
subgroupoid of $G_{m,n}$, there is a natural inclusion of their
$C^{*}$-algebras, which we also denote $j_{l}$. We define a subalgebra
$C_{l,m,n}$ of $B_{m,n}^{+}\cap C^{*}(\mathcal{F}_{l}^{+})$ as the pull-back
of these two maps, $q,j_{l}$. The inclusion coincides with our definition of
$j_{l}$. That is, we have short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C^{*}(G_{m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{C_{l,m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{j_{l}}$$\textstyle{C^{*}(H_{l,m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{l}}$$\textstyle{0}$

It is easy to check that such that $C^{*}(\mathcal{F}_{l}^{+})$ is the closure
of the union of the $C_{l,-n,n}$, over $n\geq 1$. While the terms involving
$C^{*}(H_{l,m,n})$ and $C^{*}(G_{m,n})$ are different, these $C^{*}$-algebras
are both finite dimensional and have trivial $K_{1}$-groups and this is
sufficient to conclude the inclusion of $C_{l,m,n}$ in $B_{m,n}$ induces an
isomorphism on $K_{1}$. The conclusion follows as $C^{*}(\mathcal{F}_{l}^{+})$
and $C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}}))$ are inductive limits of these
sequences. ∎

Let us continue to develop the ideas of this last proof. It is clear from the
definitions that for fixed $m,n,l$, $H_{l,m,n}$ is a subgroupoid of
$H_{l+1,m,n}$. It is also a simple matter to check that $(p,q)$ is in
$H_{l+1,m,n}$, but not in $H_{l,m,n}$ if and only if
$I(p)=I_{l+1},I(q)=I_{l^{\prime}},l^{\prime}\leq l$ or vice verse. If
$I(p)=l+1$, its equivalence class in $H_{l+1,m,n}$ consists of $q$ with
$(p,q)$ in $G_{m,n}$ and $I(q)=I_{l^{\prime}},l^{\prime}\leq l+1$ and in
$H_{l,m,n}$ this becomes two equivalence classes, those $q$ with
$I(q)=I_{l+1}$ and those $q$ with $I(q)=I_{l^{\prime}},l^{\prime}\leq l$.
Provided that such a pair $(p,q)$ exists, the map from
$K_{0}(C^{*}(H_{l,m,n}))$ to $K_{0}(C^{*}(H_{l+1,m,n})$ is surjective and has
kernel generated by $[\delta_{(p,p)}]_{0}-[\delta_{(q,q)}]_{0}$. If we
consider the exact sequences on $K$-groups associated with the commutative
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{l+1,m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C^{*}(H_{l+1,m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{v\in
V_{n}}A_{m,n,v}^{+}\otimes
C_{0}(0,\nu_{s}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{C_{l,m,n}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{i_{l+1,l}}$$\textstyle{C^{*}(H_{l,m,n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{l+1,l}}$$\textstyle{0}$

the $[\delta_{(p,p)}]_{0}-[\delta_{(q,q)}]_{0}$ is also in the kernel of the
index map and hence lifts to a non-zero class we denote by $\alpha_{l}$ in
$K_{0}(C_{l,m,n}^{+})$. As the inductive limit over $m,n$ of $C_{l,m,n}$ is
$C^{*}(\mathcal{F}_{l}^{+})$, $\alpha_{l}$ also represents a non-zero class in
$K_{0}(C^{*}(\mathcal{F}_{l}^{+}))$ which freely generates the kernel of the
map to $K_{0}(C^{*}(\mathcal{F}_{l+1}^{+}))$ induces by the inclusion.

As for the existence of the pair $(p,q)$, we know that
$i(I_{l+1})=i(I_{l^{\prime}})$, for some $l^{\prime}\in\mathcal{I}_{0}$. We
may find $(i,j)$ in $I_{\mathcal{B}}^{+}\star_{\Delta}J_{\mathcal{B}}^{+}$ and
$x,y$ in $T^{+}(x_{i})\cap\Delta_{s}(T^{+}(x_{j})$ with $\pi(x)$ in $I_{l+1}$
and $\pi(y)$ in $I_{l^{\prime}}$. The fact that $(x,y)$ is in $T^{+}$, there
exists $n\geq 1$ such that $x_{(n,\infty)}=y_{(n,\infty)}$, As $I_{l}$ and
$I_{l^{\prime}}$ are open, we may find $m<0$ such that
$\pi(X^{-}_{s(x_{m})}x_{(m,\infty)})\subseteq I_{l+1}$ while
$\pi(X^{-}_{s(y_{m})}x_{(m,\infty)})\subseteq I_{l^{\prime}}$. Letting
$p=[x_{m},x_{n}]$ and $q=[y_{m},y_{n}]$, this pair satisfies the hypotheses
for this particular $m,n$. The same argument works for all lesser $m$ and
greater $n$. We have proved the following.

######  Lemma 10.8.

For each $l\geq 1$, with $\alpha_{l}$ as above, there is a short exact
sequence

$0\rightarrow\mathbb{Z}\alpha_{l}\rightarrow
K_{0}(C^{*}(\mathcal{F}_{l}^{+}))\stackrel{{\scriptstyle(i_{l+1,l})_{*}}}{{\longrightarrow}}K_{0}(C^{*}(\mathcal{F}_{l+1}^{+}))\rightarrow
0.$

######  Theorem 10.9.

There is a short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ker(\iota)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{K_{0}(C^{*}(\mathcal{F}_{\mathcal{B}}^{+}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\textstyle{K_{0}(C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}})))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $j$ denotes the inclusion of $C^{*}(\mathcal{F}_{\mathcal{B}}^{+})$ in
$C^{*}_{r}(T^{\sharp}(S_{\mathcal{B}}))$.

###### Proof.

As we noted above, we can list a free set of generators for $\ker(\iota)$ as
follows. For each $l\geq 1$, let $(i,j)=\iota(I_{l})$. There is a unique
$I_{l}^{\prime}$ in $\mathcal{I}_{0}$ with
$\iota(I_{l})=\iota(I_{l}^{\prime})$ and $I_{l}-I_{l}^{\prime}$, as an element
of $\mathbb{Z}\mathcal{I}_{\mathcal{B}}$ and in $\ker(\iota)$. As $l\geq 1$
varies, these form a free set of generators.

We define the inclusion $\beta$ of $\ker(\iota)$ in
$K_{0}(C^{*}(\mathcal{F}_{\mathcal{B}}^{+}))$ as follows. Since each map
$(i_{l+1,l})_{*}$ is surjective, so are their compositions. So for each $l\geq
1$, we may find $\beta_{l}$ in
$K_{0}(C^{*}(\mathcal{F}_{0}^{+}))=K_{0}(C^{*}(\mathcal{F}_{\mathcal{B}}^{+}))$
such that $(i_{l,0})_{*}(\beta_{l})=\alpha_{l}$, as in Lemma 10.8. For any
integers $k_{l},1\leq l\leq L$, define

$\beta\left(\sum_{l=1}^{L}k_{l}(I_{l}-I_{l}^{\prime})\right)=\sum_{l=1}^{L}k_{l}\beta_{l}.$

Let us first observe that, if $m>l$, then

$(i_{m,0})_{*}(\beta_{l})=(i_{m,l+1})_{*}\circ(i_{l+1,l})_{*}\circ(i_{l,0})_{*}(\beta_{l})=(i_{m,l+1})_{*}\circ(i_{l+1,l})_{*}(\alpha_{l})=0.$

The fact that the image of $\beta$ is precisely the kernel of $j_{*}$ can be
seen as follows. As the union of the $C^{*}(\mathcal{F}_{l}^{+}),l\geq 0,$ is
dense in $C^{*}(T^{\sharp}_{\mathcal{B}},S_{\mathcal{B}})$,
$K_{0}(C^{*}(T^{\sharp}_{\mathcal{B}},S_{\mathcal{B}}))$ the inductive limit
of

$K_{0}(C^{*}(\mathcal{F}_{B}^{+}))=K_{0}(C^{*}(\mathcal{F}_{0}^{+}))\stackrel{{\scriptstyle(i_{1,0})_{*}}}{{\rightarrow}}K_{0}(C^{*}(\mathcal{F}_{1}^{+}))\stackrel{{\scriptstyle(i_{2,1})_{*}}}{{\rightarrow}}\cdots$

First, as each $(i_{l+1,l})_{*}$ is surjective, so is $j_{*}$. It also follows
that $a$ in $K_{0}(C^{*}(\mathcal{F}_{B}^{+}))$ is in the kernel of $j_{*}$ if
and only if $(i_{m,0})_{*}(a)=0$, for some $m\geq 1$. If
$a=\beta\left(\sum_{l=1}^{L}k_{l}(I_{l}-I_{l}^{\prime})\right)$, then this
holds for any $m>L$ from the definition of $\beta$ and our observation above.

Conversely, suppose $0=(i_{m,0})_{*}(a)$, for some $m$. This means that
$(i_{m-1,0})_{*}(a)$ is in the kernel of $(i_{m,m-1})_{*}$ and hence there is
an integer $k_{m-1}$ such that $(i_{m-1,0})_{*}(a)=k_{m-1}\alpha_{m-1}$. It
then follows that

$(i_{m-1,m-2})_{*}\circ(i_{m-2,0})_{*}(a-k_{m-1}\beta_{m-1})=(i_{m-1,0})_{*}(a-k_{m-1}\beta_{m-1})=0,$

so we may find $k_{m-2}$ such that
$(i_{m-2,0})_{*}(a-k_{m-1}\beta_{m-1})=k_{m-2}\alpha_{m-2}$. Continuing in
this way ends by seeing that $a=\sum_{l=1}^{m-1}k_{l}\beta_{l}$ as desired.

Let us finally show that $\beta$ is injective. Suppose that
$\beta\left(\sum_{l=1}^{L}k_{l}(I_{l}-I_{l}^{\prime})\right)=0$. It follows
that

$(i_{L,0})_{*}\left(\beta\left(\sum_{l=1}^{L}k_{l}(I_{l}-I_{l}^{\prime})\right)\right)=\sum_{l=1}^{L}k_{l}(i_{L,0})_{*}(\beta_{l})=k_{L}\alpha_{L},$

using again the observation above that $(i_{l,0})_{*}(\beta_{l})=0$ if $L>l$.
As $\alpha_{L}$ has infinite order, it follows that $k_{L}=0$. Continuing in
this way shows that $k_{l}=0$, for all $1\leq l\leq L$. ∎

As we indicated earlier, knowing the $K$-theory of the foliation algebra
allows the computation of the $K$-theory of the surface $S_{\mathcal{B}}$ as
an immediate consequence of Connes’ analogue of the Thom isomorpism Theorem:
10.2.2 of [Bla86].

######  Theorem 10.10.

We have $K^{i+1}(S_{\mathcal{B}})\cong
K_{i}(C^{*}(\mathcal{F}_{\mathcal{B}}^{+}))$.

######  Corollary 10.11.

Let $\mathcal{B}$ be a bi-infinite ordered Bratteli diagram satisfying the
conditions of 6.2. If $K_{0}(A_{\mathcal{B}}^{+})$ is not finitely generated
then the surface $S_{\mathcal{B}}$ has infinite genus.

##  11\. Chamanara’s surface

Figure 1. Two presentations of the Chamanara surface: the interiors of the
edges with the same label are identified by a translation. The point at the
boundary of such edges are not part of the surface and the surface has
infinite genus. The presentation on the left is the standard presentation.

There is a family of surfaces of infinite genus introduced by Chamanara
[Cha04] which kicked off the study of flat geometry and dynamics of surfaces
of infinite genus. The simplest of them has become known as _the_ Chamanara
surface (see Figure 1). Later, in [LT16], the connection was made between this
surface, the Bratteli diagram of the $2^{\infty}$ UHF $C^{*}$-algebra, and the
diadic odometer. In this section we apply our machinery to study the different
algebras and their $K$-theory.

The bi-infinite, ordered Bratteli diagram which is relevant here has the
properties

$\displaystyle V_{n}$ $\displaystyle=$ $\displaystyle\\{v_{n}\\},$
$\displaystyle E_{n}$ $\displaystyle=$ $\displaystyle\\{0_{n},1_{n}\\},$
$\displaystyle 0_{n}$ $\displaystyle\leq_{r}$ $\displaystyle 1_{n},$
$\displaystyle 0_{n}$ $\displaystyle\leq_{s}$ $\displaystyle 1_{n}$

for all $n$ in $\mathbb{Z}$. This diagram has a state which is unique, up to
scaling:

$\nu_{r}(v_{n})=2^{n},\nu_{s}(v_{n})=2^{-n},n\in\mathbb{Z}.$

It is easy to see that

$X^{ext}_{\mathcal{B}}=\\{(\cdots\,1\,1\,1\,\cdots),(\cdots\,0\,0\,0\,\cdots)\\}.$

It is also clear that in Proposition 7.4 that we have
$I_{\mathcal{B}}=J_{\mathcal{B}}$ and we can use
$x_{1}=1^{\infty}=(\cdots\,1\,1\,1\,\cdots)$ and
$x_{2}=0^{\infty}=(\cdots\,0\,0\,0\,\cdots)$. It is also easy to see that
$\partial^{s}X_{\mathcal{B}}$ consists of sequences that have a last $0$, or a
last $1$, while $\partial^{r}X_{\mathcal{B}}$ consists of sequences that have
a first $0$, or a first $1$. Among these, for each integer $n$, we define four
special points:

  * •

$w^{n}$: has a $1$ in entry $n$ and $0$’s elsewhere,

  * •

$x^{n}$: has $0$ in all entries $\leq n$ and $1$’s elsewhere,

  * •

$y^{n}$: has $1$ in all entries $\leq n$ and $0$’s elsewhere,

  * •

$z^{n}$: has a $0$ in entry $n$ and $1$’s elsewhere.

It is easy to check that

$\begin{array}[]{rclrcl}\Delta_{s}(w^{n})&=&x^{n},&\Delta_{r}(w^{n})&=&y^{n-1}\\\
\Delta_{s}(z^{n})&=&y^{n},&\Delta_{r}(z^{n})&=&x^{n-1}\end{array}$

It follows that

$\Delta_{r}\circ\Delta_{s}(w^{n})=z^{n+1}\neq
z^{n-1}=\Delta_{s}\circ\Delta_{r}(w^{n})$

and

$\\{w^{n},x^{n},y^{n},z^{n}\mid
n\in\mathbb{Z}\\}\subseteq\Sigma_{\mathcal{B}}.$

The reverse containment is quite easy.

It is fairly easy to check that the functions
$\varphi^{1^{\infty}}_{r},\varphi^{0^{\infty}}_{r}$ can be written quite
explicitly as

$\displaystyle\varphi^{1^{\infty}}_{r}(x)$ $\displaystyle=$
$\displaystyle-\sum_{n\in\mathbb{Z}}2^{n}(1-x_{n}),$
$\displaystyle\varphi^{0^{\infty}}_{r}(y)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}}2^{n}y_{n},$

for any $x$ in $T^{+}(1^{\infty})$ and $y$ in $T^{+}(0^{\infty})$,
respectively. The ranges are

$\displaystyle\varphi^{1^{\infty}}_{r}(T^{+}(1^{\infty})\cap Y_{\mathcal{B}})$
$\displaystyle=$ $\displaystyle\cup_{n\in\mathbb{Z}}(-2^{n},-2^{n+1}),$
$\displaystyle\varphi^{0^{\infty}}_{r}(T^{+}(0^{\infty})\cap Y_{\mathcal{B}})$
$\displaystyle=$ $\displaystyle\cup_{n\in\mathbb{Z}}(2^{n},2^{n+1}).$

The quotient map then identifies each interval of the former with the
corresponding interval in the latter having the same length.

Moving on to K-theory, we have
$K_{0}(A_{\mathcal{B}}^{+})\cong\mathbb{Z}[1/2]$, as ordered abelian groups,
with the latter having the usual order from the real numbers. In fact, the map
sends the class of a projection $a_{p,p},p\in E_{m,n}$ to
$2^{-n}=\nu_{s}(r(p))$.

Theorem 10.5 then tells us that $K_{0}(B_{\mathcal{B}})\cong
K_{0}(A^{+}_{\mathcal{B}})\cong\mathbb{Z}[1/2]$, as ordered abelian groups.
The collection of connected subsets of $T^{\sharp}(1^{\infty})$ is indexed by
the integers: interval $n$ having length $2^{n}$.That is, we have a canonical
identification of $\mathcal{I}_{\mathcal{B}}=\\{I_{n}\mid n\in\mathbb{Z}\\}$,
where $I_{n}$ has length $2^{n}$. The map $\iota$ of Theorem 10.9 is induced
by sending each generator $I_{n}$ to the same thing, so $\ker(\iota)$ is the
free abelian group with generators $I_{n}-I_{n-1}$.

We claim that $K_{0}(C^{*}(\mathcal{F}_{\mathcal{B}}))$ is the free abelian
group on a countably infinite set, which we will index by the integers. We
will now explicitly write a set of generators.

Fix an integer $m$ and consider $p_{m}=01011$ and $q_{m}=01010$ in
$E_{m-5,m}^{Y}$. (The presence of two $0$’s and two $1$’s guarantees that we
avoid $\Sigma_{\mathcal{B}}$.) Define

(2)
$w_{m}=\cos\left(\frac{\pi}{2}2^{-m}\varphi^{v_{m}}_{s}(z)\right)a_{p_{m},p_{m}}+\sin\left(\frac{\pi}{2}2^{-m}\varphi^{v_{m}}_{s}(z)\right)a_{q_{m},p_{m}},$

for $z$ in $X_{v_{m}}^{+}$, which lies in $AC_{m-5,m}$. A simple computation
shows $w_{m}^{*}w_{m}=a_{p_{m},p_{m}}$ while $w_{m}w_{m}^{*}$ equals
$a_{p_{m},p_{m}}$ when evaluated at $z=x_{v_{m}}^{s-min}$ and equals
$a_{q_{m},q_{m}}$ when evaluated at $z=x_{v_{m}}^{s-max}$. (This is the same
$w_{m}$ appearing in the proof of Theorem 10.5.) Just as in Theorem 10.5, this
shows that $w_{m}w^{*}_{m}$ lies in $B_{\mathcal{B}}$ and is Murray-von
Neumann equivalent to $a_{p_{m},p_{m}}$ in $A_{\mathcal{B}}^{Y+}$. In
particular, identifying $K_{0}(A^{+}_{\mathcal{B}})\cong\mathbb{Z}[1/2]$,
$j_{*}([w_{m}w_{m}^{*}])=2^{-m}$.

There is a geometric way to visualize the functions $w_{m}$ in (2). First,
since $|V_{m}|=1$ for all $m$, we have that
$X_{v_{m}}^{-}X_{v_{m}}^{+}=X_{\mathcal{B}}$ for all $m$. What the different
presentations of $X_{\mathcal{B}}$ as $X_{v_{m}}^{-}X^{+}_{v_{m}}$ highlight
are the special types of paths, e.g. $x_{v_{m}}^{s-max/min}$. This type of
different presentation is analogous to the different presentations of
Chamanara’s, e.g. the two presentations in Figure 1.

Figure 2. The functions $\bar{w}_{0}$ (in black) and $\bar{w}_{1}$ (in red) on
the two presentations of Chamanara’s surface from Figure 1. For $\bar{w}_{0}$,
white is 0, black is 1, and grey is in between, whereas for $\bar{w}_{1}$
white is 0, red is 1, and the other shades of red are in between.

Let $\pi:Y_{\mathcal{B}}\rightarrow S_{\mathcal{B}}$ be the map from the
(nonsingular) path space to the surface. The paths $p_{m},q_{m}\in
E^{Y}_{m-5,m}$ define cylinder sets $[p_{m}],[q_{m}]\subset Y_{\mathcal{B}}$
and the image of these cylinder sets under $\pi$ is denoted by
$U_{m},V_{m}\subset S_{\mathcal{B}}$. The functions $w_{m}$ in (2) are in fact
the pullback of functions $\bar{w}_{m}$ on $S_{\mathcal{B}}$ which are
supported on $U_{m}\cup V_{m}$: $w_{m}=\pi^{*}\bar{w}_{m}$, see Figure 2.

It follows then that $[w_{m}w_{m}^{*}]-2[w_{m+1}w_{m+1}^{*}]$ is in
$\ker(j_{*})$. Finally, one can show that under the identification of
$\ker(j_{*})$ with $\ker(\iota)$ given in Theorem 10.9, this element
corresponds to $I_{m}-I_{m+1}$. This computation is rather long and involves a
lot of technical details from the main results of [Put21] that we do not
provide. However, given this, it is a fairly simple matter to show that the
collection $[w_{m}w_{m}^{*}],m\in\mathbb{Z},$ generates all of
$K_{0}(C^{*}(\mathcal{F}_{\mathcal{B}}))$ and has no relations, completing the
proof of our claim above that the group is free abelian with a generating set
indexed by the integers.

##  12\. Translation surfaces of finite genus

The general goal of this section is to relate our constructions to the well-
established study of translation surfaces in the finite genus case. More
specifically, we aim to show that all finite genus translation surfaces whose
vertical and horizontal foliations are minimal arise via our construction or,
to be more precise, to see how standard techniques may be used to produce
ordered Bratteli diagrams for finite genus surfaces.

There are several equivalent ways to define a compact translation surface.
Here we give two and refer the reader to [Via06, Zor06, FM14] for thorough
introductions to flat surfaces.

Let $S$ be a compact Riemann surface of genus $g>1$ and $\alpha$ a 1-form on
$S$ which is holomorphic with respect to the complex structure on $S$. The
pair $(S,\alpha)$ defines a flat surface and a pair of transverse foliations,
the horizontal and vertical foliations, $\mathcal{F}^{\pm}$. These are the
foliations defined by the integrable distributions of the real and imaginary
parts of $\alpha$:

$\mathcal{F}^{+}=\langle\ker\Im\alpha\rangle\hskip 72.26999pt\mbox{ and
}\hskip 72.26999pt\mathcal{F}^{-}=\langle\ker\Re\alpha\rangle.$

The unit-time parametrization of these foliations are respectively the
horizontal and vertical flows $\phi_{t}^{+}$ and $\phi_{t}^{-}$.

By the Poincaré-Hopf index theorems, since $g>1$, these foliations (and the
corresponding flows) are singular; the singular points are the zeros of
$\alpha$ and these are called the singularities of $\alpha$, which are denoted
by $\Sigma$. The 1-form $\alpha$ gives $S$ a flat metric on $S\setminus\Sigma$
as follows. Let $p\in S\setminus\Sigma$ and $p^{\prime}$ in a neighborhood of
$p$. The map $p^{\prime}\mapsto\int_{p}^{p^{\prime}}\alpha\in\mathbb{C}$
defines a chart around $p$ such that the pullback of $dz$ is $\alpha$. This
gives $S\setminus\Sigma$ a flat geometry, and the reader can verify that maps
which are change of coordinates between these types of charts are of the form
$z\mapsto z+c$, justifying the use of the name translation surface. A saddle
connection is a geodesic $\gamma\subseteq S$ with respect to the flat metric
which starts and ends in $\Sigma$. More specifically, it satisfies the
property that $\partial\gamma\subseteq\Sigma$.

The geometry fails to be flat at the singular points in $\Sigma$. At these
points the local coordinate is of the form
$d\left(\frac{z^{p+1}}{p+1}\right)=dz^{p}$ for some $p\in\mathbb{N}$, called
the degree of the singularity. At a point $z\in\Sigma$ of degree $p$, the
conical angle around $z$ is $2\pi(p+1)$. If $\Sigma=\\{z_{1},\dots,z_{k}\\}$,
and the degree at $z_{i}$ is $\kappa_{i}$, then by the Gauss-Bonnet theorem we
have that $\sum_{i=1}^{k}\kappa_{i}=2g-2$. Since the holomorphic 1-form
$\alpha$ determines the geometry of the flat surface $(S,\alpha)$, it defines
its area by $\mbox{Area}(S)=\frac{i}{2}\int_{S}\alpha\wedge\bar{\alpha}$.

Another way to define a flat surface is as follows: start with a $2n$-gon
$\bar{P}\subseteq\mathbb{C}$ with the property that edges come in parallel
pairs of the same length. That is, $\bar{P}$ has edges
$\zeta_{1}^{+},\dots,\zeta_{n}^{+},\zeta_{1}^{-},\dots,\zeta_{n}^{-}$, where
$\zeta_{i}^{+}$ and $\zeta_{i}^{-}$ are parallel and of the same length. Let
$S=\bar{P}/\sim$ be the object obtained by the identifying pairs of edges
which are parallel and of the same length: $\zeta_{i}^{+}\sim\zeta_{i}^{-}$.
The holomorphic 1-form on $S$ is the unique one which pulls back as $dz$ on
$\mathbb{C}$, although it may be singular at points where different edges
meet. The points on $S$ where this happens is the singularity set $\Sigma$.
The horizontal and vertical foliations on $S$ are now seen as the horizontal
and vertical lines in $\bar{P}$. That this definition is equivalent to the one
given above is left as an exercise for the reader who has not seen this
before.

Translation surfaces come in families: all translation surfaces of genus $g$
are elements of the moduli space $\mathcal{M}_{g}$ of translation surfaces of
genus $g$. The space $\mathcal{M}_{g}$ is finite dimensional and it is
stratified into strata $\mathcal{H}(\bar{\kappa})$, where $\bar{\kappa}$
describes how many and which types of singularities the surfaces in
$\mathcal{H}(\bar{\kappa})$ are allowed to have. The stratum
$\mathcal{H}(\bar{\kappa})$ is locally modeled by
$H^{1}(S,\Sigma;\mathbb{C})$. By the remarks above,
$\mathcal{H}(\kappa_{1},\dots,\kappa_{d})\subseteq\mathcal{M}_{g}$ if and only
if $\bar{\kappa}$ satisfies $\sum\kappa_{i}=2g-2$. The Teichmüller flow is the
1-parameter family of homeomorphisms of $\mathcal{M}_{g}$, taking
$(S,\alpha)\mapsto(S,\alpha_{t})=g_{t}(S,\alpha)$, where
$\Re\alpha_{t}=e^{-t}\Re\alpha$ and $\Im\alpha_{t}=e^{t}\Im\alpha$.

In the rest of this section, we establish a way of defining an ordered, bi-
infinite Bratteli diagram $\mathcal{B}(S,\alpha)$ for a typical choice of
compact flat surface $(S,\alpha)$.

###  12.1. Veech’s zippered rectangles

Veech [Vee82] introduced a way of presenting flat surfaces as the union of
rectangles which are “zippered” on their sides. Here we review the
construction. We will follow the conventions of Viana [Via06].

Let $\mathcal{A}$ be an alphabet of size $d\geq 4$, whose elements are usually
written as $\alpha$, and
$\pi_{0},\pi_{1}:\mathcal{A}\rightarrow\\{1,\dots,d\\}$ two bijections. We
will consider examples with $\mathcal{A}=\\{A,B,C,D\\}$. We will use $\alpha$
to denote the inverses of these functions, but instead of writing
$\alpha_{\varepsilon}(i)=\pi_{\varepsilon}^{-1}(i)$, for
$\varepsilon=0,1,1\leq i\leq d$, we write $\alpha_{i}^{\varepsilon}$. These
bijections may be written conveniently as

$\pi=\left(\begin{array}[]{cccc}\alpha_{1}^{0}&\alpha_{2}^{0}&\cdots&\alpha_{d}^{0}\\\
\alpha_{1}^{1}&\alpha_{2}^{1}&\cdots&\alpha_{d}^{1}\end{array}\right),$

the top and bottom rows being ordered lists of the elements of $\mathcal{A}$.
It will always be assumed here that $\pi$ defines an irreducible permutation,
in the sense that there is no $k<d$ such that
$\pi_{1}\circ\pi_{0}^{-1}\\{1,\dots,k\\}=\\{1,\dots,k\\}$.

Figure 3. The flat surface defined by the vectors
$\zeta_{i}=(\lambda,\tau)\in\mathbb{R}^{\mathcal{A}}_{+}\times T^{+}_{\pi}$.

We will now define vectors, $(\lambda_{\alpha},\tau_{\alpha})$ in the plane,
indexed by $\alpha$ in $\mathcal{A}$. Each $\lambda_{\alpha}$ will be required
to be positive while $\tau$ satisfies

(3) $\sum_{\pi_{0}(\alpha)\leq k}\tau_{\alpha}>0\mbox{ and
}\sum_{\pi_{1}(\alpha)\leq k}\tau_{\alpha}<0,$

for all $k<d$. We let $\mathbb{R}_{+}^{\mathcal{A}}$ to denote positive
vectors and $T^{+}_{\pi}$ denote the set of all $\tau$ in
$\mathbb{R}^{\mathcal{A}}$ satisfying inequalities (3). Given
$\zeta=(\lambda,\tau)$ in$\mathbb{R}^{\mathcal{A}}_{+}\times T^{+}_{\pi}$, let
$\Gamma=\Gamma(\pi,\lambda,\tau)\subseteq\mathbb{R}^{2}$ be the curve bounded
by the concatenation of the vectors defined by $\zeta$:

$\zeta_{\alpha_{1}^{0}},\zeta_{\alpha_{2}^{0}},\cdots,\zeta_{\alpha_{d}^{0}},-\zeta_{\alpha_{d}^{1}}^{1},-\zeta_{\alpha_{d-1}^{1}}^{1},\cdots,-\zeta_{\alpha_{1}^{1}}^{1}.$

The constraints which define $T^{+}_{\pi}$ imply that about half of the
vertices of $\Gamma$ are on the upper half plane, and the other rough half on
the lower half plane. Assuming $\Gamma$ has no self-intersections111If there
are self-intersections, there is a quick fix for it., the vector $\zeta$
defines a flat surface by first defining $\zeta_{i}^{+}$ and $\zeta^{-}_{i}$
to be the corresponding edges in the concatenation above in the upper and
lower half of the plane, respectively, and then considering the interior of
$\Gamma$ and making the identifications $\zeta_{i}^{+}\sim\zeta_{i}^{-}$ on
the boundary edges (see Figure 3).

Given the data $(\pi,\lambda,\tau)$ as above, we now define the vector $h$ in
$\mathbb{R}^{\mathcal{A}}$ by

$h_{\alpha}=-\sum_{\pi_{1}(\beta)<\pi_{1}(\alpha)}\tau_{\beta}+\sum_{\pi_{0}(\beta)<\pi_{0}(\alpha)}\tau_{\beta}.$

This is more concretely expressed as $h=-\Omega_{\pi}(\tau)$, where
$\Omega_{\pi}:\mathbb{R}^{\mathcal{A}}\rightarrow\mathbb{R}^{\mathcal{A}}$ is
the matrix defined by

$\Omega_{\alpha\beta}=\left\\{\begin{array}[]{ll}+1&\mbox{ if
$\pi_{1}(\alpha)>\pi_{1}(\beta)$ and $\pi_{0}(\alpha)<\pi_{0}(\beta)$,}\\\ \
-1&\mbox{ if $\pi_{1}(\alpha)<\pi_{1}(\beta)$ and
$\pi_{0}(\alpha)>\pi_{0}(\beta)$,}\\\ 0&\mbox{ otherwise.}\end{array}\right.$

Note that, the assumption that $\tau$ is in $T^{+}_{\pi}$ implies that
$h_{\alpha}>0$ for all $\alpha$ in $\mathcal{A}$. We define the image of the
positive cone $T^{+}_{\pi}$ under $-\Omega_{\pi}$ by
$H^{+}_{\pi}=-\Omega_{\pi}(T^{+}_{\pi})$.

We now define rectangles $R_{\alpha}^{\varepsilon}$ of width
$\lambda_{\alpha}$ and height $h_{\alpha}$ by

(4)
$\begin{split}R_{\alpha}^{0}&=\left(\sum_{\pi_{0}(\beta)<\pi_{0}(\alpha)}\lambda_{\beta},\sum_{\pi_{0}(\beta)\leq\pi_{0}(\alpha)}\lambda_{\beta}\right)\times[0,h_{\alpha}]\\\
R_{\alpha}^{1}&=\left(\sum_{\pi_{1}(\beta)<\pi_{1}(\alpha)}\lambda_{\beta},\sum_{\pi_{1}(\beta)\leq\pi_{1}(\alpha)}\lambda_{\beta}\right)\times[-h_{\alpha},0].\end{split}$

along with the “zippers”

$\begin{split}Z_{\alpha}^{0}&=\left\\{\sum_{\pi_{0}(\beta)\leq\pi_{0}(\alpha)}\lambda_{\beta}\right\\}\times\left[0,\sum_{\pi_{0}(\beta)\leq\pi_{0}(\alpha)}\tau_{\beta}\right]\\\
Z_{\alpha}^{1}&=\left\\{\sum_{\pi_{1}(\beta)\leq\pi_{1}(\alpha)}\lambda_{\beta}\right\\}\times\left[\sum_{\pi_{1}(\beta)\leq\pi_{1}(\alpha)}\tau_{\beta},0\right]\\\
\end{split}$

Figure 4. The zippered rectangles for the surface in Figure 3.

which are vertical segments ending at the points of concatenation of the curve
$\Gamma$. As such, the flat surface $S(\pi,\lambda,\tau)$ can be presented as
the quotient of the closure of the union of the rectangles
$\\{R_{\alpha}^{0}\\}_{\alpha\in\mathbb{R}^{\mathcal{A}}}$ and zippers under a
relation defined on the edges of the rectangles. The genus $g$ of this surface
satisfies $2g=\mathrm{dim}\,\Omega_{\pi}(\mathbb{R}^{\mathcal{A}})$. The area
of the surface is $\mathrm{Area}(S(\pi,\lambda,\tau))=\lambda\cdot h$.
Moreover, the horizontal and vertical foliations are the obvious choices. See
Figure 4.

###  12.2. Rauzy-Veech Induction

Given a triple $(\pi,\lambda,\tau)$, where $\pi=\\{\pi_{0},\pi_{1}\\}$ is an
irreducible permutation, $\lambda$ in $\mathbb{R}^{\mathcal{A}}_{+}$ and
$\tau$ in $T^{+}_{\pi}$, we will define an operation which produces a new
triple $(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$ with the same
properties. This procedure is known as Rauzy-Veech induction, or RV induction.

First, let us describe what this procedure is meant to do geometrically, and
then we will give the details as to how it is done. Recall that from the
triple $(\pi,\lambda,\tau)$ the flat surface it defines can be presented in
zippered recangles form. The map
$\mathcal{R}(\pi,\lambda,\tau)=(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$
gives new data from which the same surface can be presented in zippered
rectangle form, except that the base one of the rectangles will be shorter and
the height of one of the rectangles will be longer. This is done by cutting
one of the rectangles $R_{\alpha}^{\varepsilon}$ into two and stacking one of
the subrectangles above or below another one of the rectangles. The choices of
the rectangles picked for this operation are determined by $(\pi,\lambda)$.

######  Remark 12.1.

It will be important to keep in mind one of the benefits of using Rauzy-Veech
induction: it allows us to understand the behavior of the leaf of the vertical
foliation on $S(\pi,\lambda,\tau)$ which emanates from the point on this
surface coming from the origin in $\mathbb{R}^{2}$. An analogous procedure for
the horizontal foliation will be described in §12.3.

First, we define $\pi^{\prime}$ and $\lambda^{\prime}$. Let
$\alpha(\varepsilon)=\pi^{-1}_{\varepsilon}(d)=\alpha_{d}^{\varepsilon}$. That
is, $\alpha(0)$ and $\alpha(1)$ are the last entries in the top and bottom
rows of $\pi$.

######  Definition 12.2.

We say that $(\pi,\lambda)$ has

$\mbox{{type 0} if $\lambda_{\alpha(0)}>\lambda_{\alpha(1)}$\hskip 50.58878pt
or\hskip 50.58878pt {type 1} if $\lambda_{\alpha(0)}<\lambda_{\alpha(1)}$}.$

If $(\pi,\lambda)$ is of type $\varepsilon\in\\{0,1\\}$ then the _winner_ is
the symbol $\alpha(\varepsilon)$ and the _loser_ is $\alpha(1-\varepsilon)$.

This makes sense as long as $\lambda_{\alpha(0)}\neq\lambda_{\alpha(1)}$, so
we will make the following assumption, to which we will return later.

######  Hypothesis 12.3.

The pair $(\pi,\lambda)$ satisfies
$\lambda_{\alpha(0)}\neq\lambda_{\alpha(1)}$.

If $(\pi,\lambda)$ has type 0, then $\pi^{\prime}$ is defined by

(5)
$\pi^{\prime}=\left(\begin{array}[]{cccccccc}\alpha_{1}^{0}&\cdots&\alpha_{k-1}^{0}&\alpha_{k}^{0}&\alpha_{k+1}^{0}&\cdots&\cdots&\alpha(0)\\\
\alpha_{1}^{1}&\cdots&\alpha_{k-1}^{1}&\alpha(0)&\alpha(1)&\alpha^{1}_{k+1}&\cdots&\alpha^{1}_{d-1}\end{array}\right),$

that is,

$\alpha_{i}^{0^{\prime}}=\alpha_{i}^{0}\hskip 28.90755pt\mbox{ and }\hskip
28.90755pt\alpha^{1^{\prime}}_{i}=\left\\{\begin{array}[]{ll}\alpha_{i}^{1}&\mbox{
if $i\leq\pi_{1}(\alpha(0))$}\\\ \alpha(1)&\mbox{ if
$i=\pi_{1}(\alpha(0))+1$}\\\ \alpha_{i-1}^{1}&\mbox{ if
$i>\pi_{1}(\alpha(0))+1$}\end{array}\right..$

The vector $\lambda^{\prime}$ is now defined by

(6) $\lambda_{\alpha}^{\prime}=\lambda_{\alpha}\mbox{ if
$\alpha\neq\alpha(0)$\;\;\; and \;\;\;
$\lambda_{\alpha(0)}^{\prime}=\lambda_{\alpha(0)}-\lambda_{\alpha(1)}$},$

whereas $\tau^{\prime}$ is defined by

(7) $\tau_{\alpha}^{\prime}=\tau_{\alpha}\mbox{ if
$\alpha\neq\alpha(0)$\;\;\;and \;\;\;
$\tau_{\alpha(0)}^{\prime}=\tau_{\alpha(0)}-\tau_{\alpha(1)}$}.$

If $(\pi,\lambda)$ has type 1, then $\pi^{\prime}$ is defined by

(8)
$\pi^{\prime}=\left(\begin{array}[]{cccccccc}\alpha_{1}^{0}&\cdots&\alpha_{k-1}^{0}&\alpha(1)&\alpha(0)&\alpha^{0}_{k+1}&\cdots&\alpha^{0}_{d-1}\\\
\alpha_{1}^{1}&\cdots&\alpha_{k-1}^{1}&\alpha_{k}^{1}&\alpha_{k+1}^{1}&\cdots&\cdots&\alpha(1)\end{array}\right),$

that is,

$\alpha_{i}^{1^{\prime}}=\alpha_{i}^{1}\hskip 28.90755pt\mbox{ and }\hskip
28.90755pt\alpha^{0^{\prime}}_{i}=\left\\{\begin{array}[]{ll}\alpha_{i}^{0}&\mbox{
if $i\leq\pi_{0}(\alpha(1))$}\\\ \alpha(0)&\mbox{ if
$i=\pi_{0}(\alpha(1))+1$}\\\ \alpha_{i-1}^{1}&\mbox{ if
$i>\pi_{0}(\alpha(1))+1$}\end{array}\right..$

The vector $\lambda^{\prime}$ is now defined by

(9) $\lambda_{\alpha}^{\prime}=\lambda_{\alpha}\mbox{ if
$\alpha\neq\alpha(1)$\;\;\; and \;\;\;
$\lambda_{\alpha(1)}^{\prime}=\lambda_{\alpha(1)}-\lambda_{\alpha(0)}$},$

whereas $\tau^{\prime}$ is defined by

(10) $\tau_{\alpha}^{\prime}=\tau_{\alpha}\mbox{ if
$\alpha\neq\alpha(1)$\;\;\;and \;\;\;
$\tau_{\alpha(1)}^{\prime}=\tau_{\alpha(1)}-\tau_{\alpha(0)}$}.$

Let
$\Theta=\Theta_{\pi,\lambda}:\mathbb{R}^{\mathcal{A}}\rightarrow\mathbb{R}^{\mathcal{A}}$
be the matrix defined, when $(\pi,\lambda)$ has type 0,as

$\Theta_{\alpha\gamma}=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\alpha=\gamma\\\ 1&\mbox{ if $\alpha=\alpha(1)$ and $\gamma=\alpha(0)$}\\\
0&\mbox{ otherwise}\end{array}\right.$

whose inverse is

$\Theta_{\alpha\gamma}^{-1}=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\alpha=\gamma\\\ -1&\mbox{ if $\alpha=\alpha(1)$ and $\gamma=\alpha(0)$}\\\
0&\mbox{ otherwise.}\end{array}\right.$

When $(\pi,\lambda)$ has type 1,

$\Theta_{\alpha\gamma}=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\alpha=\gamma\\\ 1&\mbox{ if $\alpha=\alpha(0)$ and $\gamma=\alpha(1)$}\\\
0&\mbox{ otherwise}\end{array}\right.$

whose inverse is

$\Theta_{\alpha\gamma}^{-1}=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\alpha=\gamma\\\ -1&\mbox{ if $\alpha=\alpha(0)$ and $\gamma=\alpha(1)$}\\\
0&\mbox{ otherwise.}\end{array}\right.$

This matrix satisfies [Via06, Lemma 10.2] the relation

(11) $\Theta\Omega_{\pi}\Theta^{*}=\Omega_{\pi^{\prime}}.$

As such, the relations between $\lambda$ and $\lambda^{\prime}$ and between
$\tau$ and $\tau^{\prime}$, are expressed by

$\lambda^{\prime}=\Theta^{-1*}\lambda\hskip 14.45377pt\mbox{ or }\hskip
14.45377pt\lambda=\Theta^{*}\lambda^{\prime},\hskip 21.68121pt\mbox{ and
}\hskip 21.68121pt\tau^{\prime}=\Theta^{-1*}\tau,$

and so Rauzy-Veech induction is the map

$\mathcal{R}:(\pi,\lambda,\tau)\mapsto(\pi^{\prime},\Theta^{-1*}\lambda,\Theta^{-1*}\tau).$

In terms of zippered rectangles, RV induction has an explicit expression in
terms of the height vector $h=-\Omega_{\pi}(\tau)$ in $H^{+}_{\pi}$. Indeed,
we have that $\Theta\Omega_{\pi}=\Omega_{\pi^{\prime}}\Theta^{-1*}$ and so
denoting $h^{\prime}=-\Omega_{\pi^{\prime}}(\tau^{\prime})$ the corresponding
height vector for $\tau^{\prime}$, we have that $h^{\prime}=\Theta h$. It is
straight forward to verify that if $\tau$ in $T^{+}_{\pi}$ then
$\tau^{\prime}$ in $T^{+}_{\pi^{\prime}}$. As such, the surface
$S(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$ has area

$\begin{split}\mathrm{Area}(S(\pi^{\prime},\lambda^{\prime},\tau^{\prime}))&=\lambda^{\prime}\cdot(-\Omega_{\pi^{\prime}}(\tau^{\prime}))=-\Theta^{-1*}\lambda\cdot\Theta\Omega_{\pi}\Theta^{*}(\tau^{\prime})=-\Theta^{-1*}\lambda\cdot\Theta\Omega_{\pi}(\tau)\\\
&=\Theta^{-1*}\lambda\cdot\Theta h=\lambda\cdot
h=\mathrm{Area}(S(\pi,\lambda,\tau)).\end{split}$

Figure 5. Geometric illustration of Rauzy-Veech induction. Since
$\lambda_{\alpha(1)}>\lambda_{\alpha(0)}$, this corresponds to type 1.

Geometrically, Rauzy-Veech induction makes a vertical cut through the widest
rectangle at the end, takes the right subrectangle, and stacks it above or
below the rectangle according to the rules described above. Figure 5
illustrates an example of what Rauzy-Veech induction does to the zippered
rectangles and the surface it represents from Figure 4.

Note that in the definition of RV induction, whenever it was defined
(Hypothesis 12.3), we may have that $\pi^{\prime}\neq\pi$. Thus we can
consider all possible permutations that can be obtained from $\pi$ under RV
induction.

######  Definition 12.4.

The _Rauzy graph_ $\mathcal{G}_{d}$ of permutations on $d$ elements is the
directed graph which has as vertices equivalence classes of permutations
$\pi=\\{\pi_{0},\pi_{1}\\}$, where $\pi\sim\pi^{\prime}$ whenever
$\pi_{1}\circ\pi_{0}^{-1}=\pi_{1}^{\prime}\circ\pi_{0}^{-1^{\prime}}$, and
there is an edge from $[\pi]$ to $[\pi^{\prime}]$ if there are representatives
$\pi$, $\pi^{\prime}$ and vector $\lambda$ in $\mathbb{R}^{\mathcal{A}}_{+}$
such that $\pi^{\prime}$ is the permutation obtained from $(\pi,\lambda)$
through Rauzy-Veech induction. A _Rauzy class_ is by connected components of
the Rauzy graph.

There are two outgoing edges from each class $[\pi]$, one for each type, as
well as two incoming edges. See Figures 9 and 10 for examples in genus 2.

Let $(\pi,\lambda,\tau)$ in $\mathcal{C}\times\mathbb{R}^{\mathcal{A}}\times
T^{+}_{\pi}$ satisfying Hypothesis 12.3. Then the map $\mathcal{R}$ is well
defined, and we obtain
$(\pi^{\prime},\lambda^{\prime},\tau^{\prime})=\mathcal{R}(\pi,\lambda,\tau)$
in $\mathcal{C}\times\mathbb{R}^{\mathcal{A}}\times T^{+}_{\pi}$. We would
like to once again apply $\mathcal{R}$ to this new data, but we do not know
a-priori whether $(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$ satisfies
Hypothesis 12.3.

To establish conditions for which all iterates of RV induction are defined, we
first need to define the interval exchange transformation (IET) defined by
$(\pi,\lambda)$. For $\alpha$ in $\mathcal{A}$, let

$I_{\alpha}=\left[\sum_{\pi_{0}(\gamma)<\pi_{0}(\alpha)}\lambda_{\gamma},\sum_{\pi_{0}(\gamma)\leq\pi_{0}(\alpha)}\lambda_{\gamma}\right)$

and with $|I|=\|\lambda\|_{1}$, the IET $f:[0,|I|)\rightarrow[0,|I|)$ defined
by $(\pi,\lambda)$ is

$f(x)=x+\Omega_{\pi}(\lambda)_{\alpha}\mbox{ for $x\in I_{\alpha}$}.$

The reader is encouraged now to verify that the zippered rectangle surfaces in
§12.1 are suspensions over the IET $f$ with roof functions given by the height
vector $h$. Denote by $\partial I_{\alpha}$ the left endpoint of the interval
$I_{\alpha}$.

######  Definition 12.5.

A pair $(\pi,\lambda)$ satisfies the _Keane condition_ if $f^{m}(\partial
I_{\alpha})\neq\partial I_{\gamma}$ for all $m$ in $\mathbb{N}$ and
$\alpha,\gamma\in\mathcal{A}$ with $\pi_{0}(\gamma)\neq 1$.

######  Remark 12.6.

  1. (1)

This condition guarantees that the orbits of the left endpoints of the
intervals are as disjoint as possible. This surely guarantees Hypothesis 12.3.
Below we will see that this characterizes the good data for which RV induction
is defined for all iterates.

  2. (2)

It is known that the Keane condition implies the minimality of the interval
exchange transformation $f$, that is, that every orbit is dense. This in turn
implies that the vertical foliation on $S(\pi,\lambda,\tau)$ has no closed
leaves and every leaf is dense in the surface.

######  Theorem 12.7.

The following are equivalent:

  1. (1)

$(\pi,\lambda)$ satisfies the Keane condition.

  2. (2)

All iterates $\mathcal{R}^{n}(\pi,\lambda)$ of Rauzy-Veech induction are
defined for $n>0$.

  3. (3)

For each $\alpha$ in $\mathcal{A}$, there is a subsequence
$n_{i}^{\alpha}\rightarrow\infty$ such that $\alpha$ is the winner for
$\mathcal{R}^{n_{i}^{\alpha}}(\pi,\lambda)$ for every $i$.

  4. (4)

For each $\alpha$ in $\mathcal{A}$, there is a subsequence
$n_{i}^{\alpha}\rightarrow\infty$ such that $\alpha$ is the loser for
$\mathcal{R}^{n_{i}^{\alpha}}(\pi,\lambda)$ for every $i$.

Moreover, these equivalent conditions are satisfied on a full measure subset
of the space of parameters.

For a proof, see [Via06, §5]. Thus, it is better to replace Hypothesis 12.3
with the Keane condition.

###  12.3. RH Induction

The previous section reviewed a procedure which, starting with some data
$(\pi,\lambda,\tau)$ and depending only on $\pi$ and $\lambda$, produced a new
triple $\mathcal{R}(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$. Moreover,
there is a precise condition that characterizes all data for which all
iterates of this procedure are defined. Denoting by
$(\pi^{(n)},\lambda^{(n)},\tau^{(n)})=\mathcal{R}^{n}(\pi,\lambda,\tau)$, the
surfaces $S(\pi^{(n)},\lambda^{(n)},\tau^{(n)})$ are different presentations
of $S(\pi,\lambda,\tau)$ which allow us to keep track of longer and longer
segments of the vertical leaf emanating from the origin.

In this section, we define a different procedure,
$\mathcal{P}:(\pi,\lambda,\tau)\mapsto(\pi^{\prime},\lambda^{\prime},\tau^{\prime})$,
with the aim of doing the same for the trajectory of the horizontal foliation
emanating from the origin, that is, we will get presentations
$S(\pi^{(n)},\lambda^{(n)},\tau^{(n)})$ of $S(\pi,\lambda,\tau)$ through
$(\pi^{(n)},\lambda^{(n)},\tau^{(n)})=\mathcal{P}^{n}(\pi,\lambda,\tau)$ which
will allow us to capture longer and longer segments of the horizontal leaf
emanating from the origin. We will then relate this procedure to RV induction.
This exposition is our own, but the recent work [Ber21] captures most of the
aspects presented here.

Let us first describe and illustrate how this procedure is meant to work and
then we will give the details. Let $(\pi,\lambda,\tau)$ in
$\mathcal{C}\times\mathbb{R}^{\mathcal{A}}_{+}\times T^{+}_{\pi}$ and consider
the zippered rectangles presentation of it in (4). Our goal is to extend
$[0,|I|)$ to $[0,|I^{\prime}|)$, where
$|I^{\prime}|=\|\lambda^{\prime}\|_{1}$. Given $\pi=(\pi_{0},\pi_{1})$, define

(12)
$\beta(\varepsilon)=\pi_{\varepsilon}^{-1}(\pi_{\varepsilon}(\alpha(1-\varepsilon))+1),$

for $\varepsilon=\\{0,1\\}$. In words, $\beta(\varepsilon)$ is the symbol
immediately to the right of $\alpha_{d}^{1-\varepsilon}$ on
$\pi_{\varepsilon}$.

Figure 6. The case $h_{\alpha(1)}>h_{\beta(0)}$ and
$h_{\alpha(0)}<h_{\beta(1)}$.

Suppose for the moment that $h_{\alpha(1)}>h_{\beta(0)}$ and
$h_{\alpha(0)}<h_{\beta(1)}$ (see Figure 6). In order to extend the horizontal
leaf starting at the origin, it must come out of the bottom edge of the
rectangle $R^{1}_{\alpha(0)}$ cut through $R^{1}_{\beta(1)}$, subdividing it
into two subrectangles. The top rectangle will be absorbed into a larger
rectangle $R^{1}_{\alpha^{\prime}(0)}=R^{1}_{\alpha(0)}$, while the bottom
rectangle will be moved to the right and become $R^{1}_{\alpha^{\prime}(1)}$.
Thus, we extend $[0,|I|)$ by $\lambda_{\beta(1)}$ and rearrange the rectangles
as in Figure 7.

If $h_{\alpha(1)}<h_{\beta(0)}$ and $h_{\alpha(0)}>h_{\beta(1)}$ then an
analogous procedure is defined by cutting through the rectangle
$R^{0}_{\beta(0)}$ and moving the top rectangle to the right-most place on the
top set of rectangles.

It may be unclear how to proceed if $h_{\alpha(1)}<h_{\beta(0)}$ and
$h_{\alpha(0)}<h_{\beta(1)}$, as in Figure 4. What really determines which
rectangle to cut has to do with the $\tau\in T^{+}_{\pi}$ which defines
$h=-\Omega_{\pi}(\tau)$. Indeed, in the case $h_{\alpha(1)}>h_{\beta(0)}$ and
$h_{\alpha(0)}<h_{\beta(1)}$ as in Figure 6 the zipper between
$R^{0}_{\alpha(1)}$ and $R^{0}_{\beta(0)}$ is somewhere in the interior of the
right edge of $R^{0}_{\alpha(1)}$, meaning that it is on the right edge of
$R^{1}_{\alpha(1)}$, meaning that $\sum_{\alpha}\tau_{\alpha}<0$. Likewise,
$h_{\alpha(1)}<h_{\beta(0)}$ and $h_{\alpha(0)}>h_{\beta(1)}$ imply that
$\sum_{\alpha}\tau_{\alpha}>0$.

Let us remark that the case $h_{\alpha(1)}>h_{\beta(0)}$ and
$h_{\alpha(0)}>h_{\beta(1)}$ is impossible. Indeed, consider the zipper
between $R^{0}_{\alpha(1)}$ and $R^{0}_{\beta(0)}$. There is a singularity of
the flat surface somewhere between these two rectangles. But this singularity
is to the right of $R^{0}_{\alpha(1)}$, which means that there is a
singularity on the right edge of $R^{1}_{\alpha(1)}$, which has to have height
$\sum_{\alpha}\tau_{\alpha}<0$. The same argument for the rectangles
$R^{1}_{\alpha(0)}$ and $R^{1}_{\beta(1)}$ implies that
$\sum_{\alpha}\tau_{\alpha}>0$. Since $\tau\in T^{+}_{\pi}$, it satisfies one
of the two conditions of (3), so it is impossible to have
$h_{\alpha(1)}>h_{\beta(0)}$ and $h_{\alpha(0)}>h_{\beta(1)}$.

######  Hypothesis 12.8.

The pair $(\pi,\tau)$ with $\tau$ in $T^{+}_{\pi}$ satisfies
$\sum_{\alpha}\tau_{\alpha}\neq 0$.

Motivated by this discussion and following the terminology [Via06, §12], we
have the following definition.

######  Definition 12.9.

If the pair $(\pi,\tau)$ satisfies Hypothesis 12.8, it will be called

$\mbox{{Type $0$} if $\displaystyle\sum_{\alpha}\tau_{\alpha}>0$\ \hskip
50.58878pt or \hskip 50.58878pt {Type $1$} if
$\displaystyle\sum_{\alpha}\tau_{\alpha}<0$}.$

If $(\pi,\tau)$ is of type $\varepsilon\in\\{0,1\\}$ then the $\tau$-_winner_
is the symbol $\alpha(1-\varepsilon)$.

Thus if $(\pi,\tau)$ is type $0$, then the rectangle $R^{0}_{\beta(0)}$ will
be subdivided into two rectangles, the bottom part will be absorbed into
$R^{0}_{\alpha(1)}$ while the top part will be moved to the right to become
$R_{d}^{0^{\prime}}$. If $(\pi,\tau)$ is type $1$, the rectangle
$R^{1}_{\beta(1)}$ will be subdivided into two rectangles, the top part will
be absorbed into $R^{1}_{\alpha(0)}$ while the top part will be moved to the
right to become $R_{d}^{1^{\prime}}$, as depicted in Figure 7.

Figure 7. Starting from Figure 6, the procedure producing a new zippered
rectangles presentation implies that $\alpha^{\prime}(0)=\alpha(0)$ and
$\alpha^{\prime}(1)=\beta(1)$. Compare with Figure 5.

These operations are formally defined as follows. Let $\tau$ in $T^{+}_{\pi}$
be of type $0$. Starting with $(\pi,\lambda,\tau)$ and based on the
description in the previous paragraph, the new data
$(\pi^{\prime},\lambda^{\prime},h^{\prime})=\mathcal{P}(\pi,\lambda,h)=\mathcal{P}(\pi,\lambda,-\Omega_{\pi}(\tau))$
is defined, first, by letting

(13)
$\pi^{\prime}=\left(\begin{array}[]{cccccccc}\alpha_{1}^{0}&\cdots&\alpha_{k-1}^{0}&\alpha(1)&\alpha^{0}_{k+2}&\cdots&\alpha(0)&\beta(0)\\\
\alpha_{1}^{1}&\cdots&\alpha_{k-1}^{1}&\alpha_{k}^{1}&\alpha_{k+1}^{1}&\cdots&\cdots&\alpha(1)\end{array}\right),$

that is,

$(\alpha^{{}^{\prime}})^{1}_{i}=\alpha_{i}^{1}\hskip 28.90755pt\mbox{ and
}\hskip
28.90755pt(\alpha^{{}^{\prime}})^{0}_{i}=\left\\{\begin{array}[]{ll}\alpha_{i}^{0}&\mbox{
if $i\leq\pi_{0}(\alpha(1))$}\\\ \alpha^{0}_{i+1}&\mbox{ if
$\pi_{0}(\alpha(1))<i<d$}\\\ \beta(0)&\mbox{ if $i=d$}\end{array}\right..$

The vector $\lambda^{\prime}$ is now defined by

(14) $\lambda_{\alpha}^{\prime}=\lambda_{\alpha}\mbox{ if
$\alpha\neq\alpha(1)$\;\;\; and \;\;\;
$\lambda_{\alpha(1)}^{\prime}=\lambda_{\alpha(1)}+\lambda_{\beta(0)}$},$

whereas $h^{\prime}$ is defined by

(15) $h_{\alpha}^{\prime}=h_{\alpha}\mbox{ if $\alpha\neq\beta(0)$\;\;\;and
\;\;\; $h_{\beta(0)}^{\prime}=h_{\beta(0)}-h_{\alpha(1)}$}.$

The definition of $\tau^{\prime}$ will follow from Proposition 12.9.

Let $\tau$ in $T^{+}_{\pi}$ be of type $1$. Starting from $(\pi,\lambda,\tau)$
we now define $(\pi^{\prime},\lambda^{\prime},h^{\prime})$ by

(16)
$\pi^{\prime}=\left(\begin{array}[]{cccccccc}\alpha_{1}^{0}&\cdots&\alpha_{k-1}^{0}&\alpha_{k}^{0}&\alpha_{k+1}^{0}&\cdots&\cdots&\alpha(0)\\\
\alpha_{1}^{1}&\cdots&\alpha_{k-1}^{1}&\alpha(0)&\alpha^{1}_{k+2}&\cdots&\alpha(1)&\beta(1)\end{array}\right),$

that is,

$\alpha_{i}^{0^{\prime}}=\alpha_{i}^{0}\hskip 28.90755pt\mbox{ and }\hskip
28.90755pt\alpha^{1^{\prime}}_{i}=\left\\{\begin{array}[]{ll}\alpha_{i}^{1}&\mbox{
if $i\leq\pi_{1}(\alpha(0))$}\\\ \alpha^{1}_{i+1}&\mbox{ if
$\pi_{1}(\alpha(0))<i<d$}\\\ \beta(1)&\mbox{ if $i=d$}\end{array}\right..$

The vector $\lambda^{\prime}$ is now defined by

(17) $\lambda_{\alpha}^{\prime}=\lambda_{\alpha}\mbox{ if
$\alpha\neq\alpha(0)$\;\;\; and \;\;\;
$\lambda_{\alpha(0)}^{\prime}=\lambda_{\alpha(0)}+\lambda_{\beta(1)}$},$

whereas $h^{\prime}$ is defined by