# A review of the Elliott program of classification of simple amenable
$C^{*}$-algebras

Guihua Gong, Huaxin Lin, and Zhuang Niu

###### Abstract

We give a brief survey of the development of the Elliott program of
classification of separable simple amenable $C^{*}$-algebras.

##  1 Introduction

By a $C^{*}$-algebra, we mean a norm closed and adjoint closed subalgebra of
the algebra $B(H)$ of all bounded linear operators on some Hilbert space $H.$
There is an abstract definition of Gelfand and Naimark ([32]) for
$C^{*}$-algebras, namely, a Banach algebra with an involutive $*$-operation
such that $\|x^{*}x\|=\|x\|^{2}$ for all $x\in A.$ Let $X$ be a compact metric
space, $G$ be a group, ${\alpha}$ be a homomorphism from $G$ to the group of
homeomorphisms of $X$ and $\mu$ be a faithful ${\alpha}$-invariant Borel
probability measure. We have a representation $\pi:G\to U(B(H)),$ the unitary
group of $B(H),$ where $H=L^{2}(X,\mu).$ For each $f\in C(X),$ we have a
bounded operator $M_{f}$ defined by $M_{f}(g)=fg$ for all $g\in L^{2}(X,\mu).$
Then we obtain a $C^{*}$-algebra $C(X)\rtimes_{\alpha}G,$ the $C^{*}$-algebra
generated by $\pi(G)$ and $\\{M_{f}:f\in C(X)\\}$ as a $C^{*}$-subalgebra of
$B(H).$ This $C^{*}$-algebra is often used to study the dynamical system
$(X,G,{\alpha}).$ $C^{*}$-algebras may also arise from non-commutative
geometry, number theory and group representations, to name a few related
fields. Needless to say, it is of paramount importance to determine the
structure of a given $C^{*}$-algebra from a few known and possibly computable
data and to recognize that some $C^{*}$-algebras from different subjects may
be isomorphic as $C^{*}$-algebras. This leads to the question of
classification of $C^{*}$-algebras.

In the First International Congress of Basic Science, David Mumford gave the
opening plenary lecture on Consciousness, robots and DNA ([83]). At the very
beginning, he cautioned the audience that there would be no mathematics in his
talk. Nevertheless, by the very end of this magnificent lecture, when he ended
his talk on DNA being a measuring instrument opens a Pandora’s box, he
proposed a problem in $C^{*}$-algebras about non-commuting self-adjoint
operators to be approximated by commuting ones, which is closely related to
the Elliott program of classification of amenable $C^{*}$-algebras (see [60]).

Classification of amenable $C^{*}$-algebras might begin with Glimm’s
classification of UHF-algebras (by supernatural numbers) in the late 1950’s.
This was followed by Dixmier’s classification of matroid $C^{*}$-algebras in
the 1960’s. Ola Bratteli used the Bratteli diagram to classify AF-algebras,
i.e., $C^{*}$-algebras which are inductive limits of finite dimensional
$C^{*}$-algebras. George A. Elliott then (1976) used dimension groups to
classify AF-algebras ([14]). It was later realized that the dimension groups
are in fact the $K_{0}$-groups. Even though $K_{1}$-groups of AF-algebras are
trivial, realizing dimension groups are $K_{0}$-groups is very important
since, immediately, $K_{1}$-groups enter the picture.

By early 1990, Elliott proposed to classify simple $C^{*}$-algebras by their
K-theory.

As suggested by Elliott ([18]), separable simple $C^{*}$-algebras $A$ are
naturally divided into three cases according to their $K_{0}$-groups:
Case 1: $K_{0}(A)\not=\\{0\\}$ is a (directed) ordered group,
Case 2. $K_{0}(A)_{+}=\\{0\\}=V(A)$,
Case 3: $K_{0}(A)=K_{0}(A)_{+}$, and $V(A)\not=\\{0\\}$,
where $V(A)$ is the set of Murry-von Neumann equivalence classes of
projections in the stabilization $A\otimes{\cal K}.$

For Case 3, i.e., the case of purely infinite simple $C^{*}$-algebras, the
classification by the Elliott invariant was accomplished in the last century
by the work of Kirchberg and Phillips (see [51] and [93]) with some earlier
works of the M. Rørdam and others (cf. [98], [99], [79], [30] and [100]). This
report will only discuss the finite case.

To clarify what classification of a class of $C^{*}$-algebras by the Elliott
invariant is about, let us end this section with the following statement.

For a complete classification of a class of $C^{*}$-algebras by the Elliott
invariant, there must be three ingredients:

(1) A definition of invariants for $C^{*}$-algebras in the class; (2) A
(range) theorem that says exactly what values of the invariants defined in (1)
occur for $C^{*}$-algebras in the class; (3) the isomorphism theorem, i.e.,
the statement that any two $C^{*}$-algebras in the class with the same, i.e.,
isomorphic invariants are isomorphic.

##  2 The beginning

At the beginning of the 1990’s, Elliott presented the following classification
results:

######  Theorem 2.1 (G. A. Elliott [16]).

Let $A=\lim_{n\to\infty}A_{n}$ and $B=\lim_{n\to\infty}B_{n}$ be unital simple
$C^{*}$-algebras with real rank zero (see 2.3 below), where $A_{n}$ and
$B_{n}$ are finite direct sums of $C^{*}$-algebras of the form
$M_{r(n,i)}(C(\mathbb{T})),$ i.e., matrices over the continuous functions on
the unit circle.

Then $A\cong B$ if and only if

$(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A))\cong(K_{0}(B),K_{0}(B)_{+},[1_{B}],K_{1}(B)).$

######  Theorem 2.2 (G. A. Elliott [16]).

Let $G_{0}$ be a simple countable (ordered) dimension group, not equal to
$\mathbb{Z},$ with an order unit $u,$ and let $G_{1}$ be a countable abelian
group. It follows that there is a unital simple $C^{*}$-algebra of real rank
zero which is the inductive limit of a sequence of finite direct sums of
circle algebras such that

$(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A))=(G_{0},(G_{0})_{+},u,G_{1}).$

This is the beginning of an age that classes of simple $C^{*}$-algebras became
classifiable.

######  Definition 2.3.

Let $A$ be a $C^{*}$-algebra, $S\subset A$ be a subset, and $x\in A$ be an
element. Let $\varepsilon>0.$ We write $x\in_{\varepsilon}S,$ if there is
$s\in S$ such that $\|x-s\|<\varepsilon.$

Denote by $T(A)$ the space of all tracial states of $A.$ Denote by
$\operatorname{Aff}(T(A))$ the space of all real affine continuous functions
on $T(A).$

A unital $C^{*}$-algebra $A$ is said to have stable rank one, if the set of
invertible elements of $A$ is dense in $A.$ A non-unital $C^{*}$-algebra has
stable rank one if $\tilde{A},$ the minimum unitization of $A,$ has stable
rank one.

A unital $C^{*}$-algebra $A$ is said to have real rank zero, if the set of
invertible self-adjoint elements of $A$ is dense in $A_{s.a.},$ the self-
adjoint part of $A.$ A non-unital $C^{*}$-algebra has real rank zero, if
$\tilde{A}$ has real rank zero.

A unital commutative $C^{*}$-algebra $A=C(X),$ where $X$ is a compact
Hausdorff space, has stable rank one if and only if ${\rm dim}X\leq 1,$ and
$A$ has real rank zero, if and only if ${\rm dim}X=0.$ Every von Neumann
algebra has real rank zero. Every AF-algebra has real rank zero. It was proved
by S. Zhang [116] that every purely infinite simple $C^{*}$-algebra has real
rank zero. It is then natural to begin the classification program with simple
$C^{*}$-algebras of real rank zero in mind such as Theorem 2.1.

######  Definition 2.4.

Denote by ${\cal I}_{0}$ the class of all finite dimensional $C^{*}$-algebras.
For $k\in\mathbb{N},$ denote by ${\cal I}_{k}$ the class of $C^{*}$-algebras
which are finite direct sums of $C^{*}$-algebras with the form
$PM_{r}(C(X))P,$ where $X$ is a finite CW complex with ${\rm dim}(X)\leq k$
and $P\in M_{r}(C(X))$ is a projection.

A $C^{*}$-algebra $A$ is said to be an AH-algebra if
$A=\lim_{n\to\infty}A_{n},$ where each $A_{n}\in{\cal I}_{k(n)}$ for some
$k(n)\in\mathbb{N}.$

If there is $k\in\mathbb{N}$ such that $k(n)\leq k,$ we say that the AH-
algebra $A$ has no dimension growth.

A unital simple $C^{*}$-algebra is said to be of slow dimension growth, if
$A=\lim_{n\to\infty}A_{n},$ where
$A_{n}=\bigoplus_{i=1}^{k(n)}P_{n,j}M_{r(n,j)}(C(X_{n,j})P_{n,j}$ and
$P_{n,j}M_{r(n,j)}(C(X_{n,j})P_{n,j}\in\bigcup_{k}{\cal I}_{k}$ such that

$\displaystyle\limsup_{n\to\infty}\max_{1\leq j\leq k(n)}{{\rm
dim}(X_{n,j})\over{{\rm rank}(P_{n,j})+1}}=0.$ (e 2.1)

Let $I_{k}=\\{f\in M_{k}(C[0,1]):~{}f(0)=\lambda_{0}\cdot
1_{k},f(1)=\lambda_{1}\cdot 1_{k},\lambda_{0},\lambda_{1}\in\mathbb{C}\\}$. A
$C^{*}$-algebra with the form $M_{n}(I_{k})$ is called a dimension drop
$C^{*}$-algebras.

A $C^{*}$-algebra is said to be sub-homogeneous, if it is a $C^{*}$-subalgebra
of a $C^{*}$-algebra in ${\cal I}_{k}$ for some $k\in\mathbb{N}.$ A
$C^{*}$-algebra is said to be an ASH-algebra if it is an inductive limit of
sub-homogeneous $C^{*}$-algebras.

######  Definition 2.5.

Let $A$ be a unital separable simple $C^{*}$-algebra with tracial state space
$T(A).$ For each $\tau\in T(A),$ we extend it to a densely defined trace
$\tau\otimes Tr$ on $A\otimes{\cal K},$ where $Tr$ is the densely defined
lower semi-continuous trace on ${\cal K},$ and ${\cal K}$ is the
$C^{*}$-algebra of compact operators on $l^{2}.$ We continue to use $\tau$ for
$\tau\otimes Tr.$ Denote by $\rho_{A}:K_{0}(A)\to\operatorname{Aff}(T(A))$ the
pairing defined by $\rho_{A}([p])=\tau(p)$ for all projections $p\in
A\otimes{\cal K}$ and $\tau\in T(A).$

Let $A$ be a unital simple exact $C^{*}$-algebra and $p,q\in A\otimes{\cal K}$
be projections. We say that $A$ has strict comparison for projections if
$\tau(p)>\tau(q)$ for all $\tau\in T(A)$ implies that $q\lesssim p.$111We
avoid the discussion of quasitraces here.

The following theorem is a combination of proofs in [13] and [4] (see also
(iii) of Theorem 7.3.9 of [2]).

######  Theorem 2.6 (cf. Dădărlat-Nagy-Némethi-Pasnicu [13] and Blackadar-
Dădărlat-Rørdam [4]).

A unital simple AH-algebra with slow dimension growth has stable rank one and
has weakly unperforated $K_{0}(A).$

After Elliott’s theorem 2.1 of classification of unital simple AH algebras
with real rank zero whose building blocks have spectrum $S^{1}$, Hongbing Su
generalized Elliott’s results to certain ASH inductive limits with sub-
homogeneous building blocks whose spectra are locally one dimensional but non-
Hausdorff spaces ([104]). However, to deal with AH inductive limit systems
with higher dimensional spectrum, one faced the difficulty of dealing with
building blocks which are not semi-projective. A method to break the spectrum
and reduce the dimension (see [59], [61] and [24]) was developed. Such a
method involves decomposing elements into two parts with the major parts being
approximated by elements in a simpler $C^{*}$-algebra such as a finite
dimensional $C^{*}$-algebra, and plays an important role in the classification
theory which eventually led to the notion of tracial rank and generalized
tracial rank.

This approach led to an important early breakthrough in the Elliott program.

######  Theorem 2.7 (Elliott-Gong, Dadarlat-Gong [22], [11], [33] and [12]).

Let $A$ and $B$ be unital simple AH-algebras with no dimension growth of real
rank zero. Then $A\cong B$ if and only if

$\displaystyle(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A))\cong(K_{0}(B),K_{0}(B)_{+},[1_{B}],K_{1}(B)).$
(e 2.2)

######  Definition 2.8.

By an ordered group, we mean an abelian group which is (partially) ordered.
Let $G$ be an ordered group. An element $u\in G$ is said to be an order unit
if, for any $g\in G,$ there is $n\in\mathbb{N}$ such that $g\leq nu.$ An
ordered group is said to be simple if every $g\in(G)_{+}\setminus\\{0\\}$ is
an order unit. A simple ordered group $G$ is said to be weakly unperforated
if, for any $x\in G,$ $mx\in G^{+}\setminus\\{0\\}$ for some integer
$m\in\mathbb{N}$ implies $x\in(G)_{+}\setminus\\{0\\}.$ Denote by $S(G,u)$ the
state space of $(G,u),$ i.e., the set of all order preserving homomorphisms
$s:G\to\mathbb{R}$ such that $s(u)=1.$ It is a convex set.

Denote by $G_{tor}$ the torsion subgroup of $G.$

For any convex set $S,$ denote by $\partial_{e}(S)$ the set of extreme points
of $S.$

######  Definition 2.9.

Let $A$ be a unital separable simple $C^{*}$-algebra with tracial state space
$T(A).$ Recall that $S(K_{0}(A),[1_{A}])$ is the state space of $K_{0}(A).$
The map $\rho_{A}:K_{0}(A)\to\operatorname{Aff}(T(A))$ induces an affine map
$\rho_{A}^{\sharp}:T(A)\to S(K_{0}(A),[1_{A}])$ by
$\rho_{A}^{\sharp}(\tau)(x)=\rho_{A}(c)(\tau)$ for all $\tau\in T(A)$ and
$x\in K_{0}(A).$

Let us denote by ${\rm Ell}(A)$ the sextuple

$(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A),T(A),\rho_{A}).$

Suppose that $B$ is another unital separable simple $C^{*}$-algebra. We write

$(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A),T(A),\rho_{A})\cong(K_{0}(B),K_{0}(B)_{+},[1_{B}],K_{1}(B),T(B),\rho_{B}),$

if there is an ordered group isomorphism $\kappa_{0}:K_{0}(A)\to K_{0}(B)$
with $\kappa_{0}([1_{A}])=[1_{B}],$ a group isomorphism
$\kappa_{1}:K_{1}(A)\to K_{1}(B)$ and an affine homeomorphism
$\kappa_{T}:T(A)\to T(B)$ such that
$\rho_{B}(\kappa_{0}(x))(\tau)=\rho_{A}(x)(\kappa_{T}^{-1}(\tau))$ for all
$x\in K_{0}(A)$ and $\tau\in T(B).$ If we agree that $K_{0}(A)_{+}=\\{x\in
K_{0}(A):\rho_{A}(x)>0\\}\cup\\{0\\},$ then, we may write

${\rm Ell}(A)=((K_{0}(A),[1_{A}],T(A),\rho_{A}),K_{1}(A)).$

Based on the [17], [19], [85], [55], [56], [57], [58], a further breakthrough
was made by the following theorem:

######  Theorem 2.10 (Elliott-Gong-Li [34] and [23]).

Let $A$ and $B$ be unital simple $C^{*}$-algebras. Suppose that both $A$ and
$B$ are unital AH-algebras with no dimension growth. Then $A\cong B$ if and
only if ${\rm Ell}(A)\cong{\rm Ell}(B).$

Theorem 2.10 is made more important by the following range theorem:

######  Theorem 2.11 (Villadsen [110]).

Suppose that $G$ is a countable simple ordered group which is weakly
unperforated with Riesz interpolation such that $G/G_{tor}$ is noncyclic,
$u\in G^{+},$ $H$ is a countable abelian group, $\Delta$ is a metrizable
Choquet simplex, and $\rho:G\to\operatorname{Aff}(\Delta)$ is an order
preserving homomorphism such that
$\rho^{\sharp}(\partial_{e}(\Delta))=\partial_{e}(S(G,u)).$

Then there exists a unital simple $AH$-algebra $A=\lim_{n\to\infty}A_{n},$
where each $A_{n}\in{\cal I}_{3},$ such that ${\rm
Ell}(A)=((G,u,H),\Delta,\rho).$

It should be noted that $\mathbb{Z}\oplus G_{0},$ where $G_{0}$ is a torsion
group, with $(\mathbb{Z}\oplus G_{0})_{+}=\\{(n,g):n\in\mathbb{N},\,g\in
G_{0}\\}\cup\\{(0,0)\\}$ is weakly unperforated but (for two reasons) excluded
from the range described in Theorem 2.11.

Fortunately for the further development of classification theory, Jiang and Su
provided the following Jiang-Su algebra ${\cal Z}.$ In particular, ${\cal Z}$
is not a unital simple AH-algebra.

######  Theorem 2.12 (Jiang-Su [50]).

There is an infinite dimensional simple $C^{*}$-algebra ${\cal Z}$ which is an
inductive limit of dimension drop algebras such that

${\rm Ell}({\cal Z})={\rm Ell}(\mathbb{C}).$

######  Theorem 2.13 (Gong-Jiang-Su [39]).

For any unital simple $C^{*}$-algebra $A$, $(K_{0}(A\otimes{\cal
Z}),K_{0}(A\otimes{\cal Z})_{+})$ is weakly unperforated. Furthermore,

$\displaystyle{\rm Ell}(A\otimes{\cal Z})\cong{\rm Ell}(A)$ (e 2.3)

if and only if $(K_{0}(A),K_{0}(A)_{+})$ is weakly unperforated. In
particular, there are unital simple AH algebras $A$ (such as the Villadsen
algebra [111]) which are not ${\cal Z}$-stable.

Next let us now introduce the $C^{*}$-algebras of UCT class.

######  Definition 2.14 (Rosenberg and Schochet [102]).

A separable $C^{*}$-algebra $A$ is said to be in the UCT class, if it
satisfies the following Universal Coefficient Theorem: For any $\sigma$-unital
$C^{*}$-algebra $B,$ one has the following short exact sequence:

$\displaystyle 0\to{\rm
ext}_{\mathbb{Z}}^{1}(K_{*}(A),K_{*}(B))\,{\stackrel{{\scriptstyle\delta}}{{\to}}}\,KK^{*}(A,B)\,{\stackrel{{\scriptstyle\gamma}}{{\to}}}\,{\rm
Hom}(K_{*}(A),K_{*}(B))\to 0,$ (e 2.4)

where $\gamma$ has degree $0$ and $\delta$ has degree $1.$ The sequence is
natural in each variable and splits unnaturally.

Let ${\cal N}$ be the smallest class of separable amenable $C^{*}$-algebras
such that (1) $\mathbb{C}\in{\cal N};$ (2) ${\cal N}$ is closed under
countable inductive limits, (3) If one has the short exact sequence $0\to A\to
E\to B\to 0,$ and two of the terms are in ${\cal N},$ then so is the third;
(4) ${\cal N}$ is closed under $KK$-equivalence (see Definition 19.1.1 of [1]
for the definition).

It is known that ${\cal N}$ is closed under stable isomorphism (by (4) above)
and contains all inductive limits of type $I$ $C^{*}$-algebras (hence all ASH-
algebras). It is closed under tensor products and crossed products by
$\mathbb{R}$ and $\mathbb{Z}.$ In particular, $O_{n},O_{\infty}\in{\cal N}.$

######  Theorem 2.15 (Rosenberg and Schochet [102]).

Every separable $C^{*}$-algebra in ${\cal N}$ is in the UCT class.

Since the Elliott invariant includes $K$-theory, one expects that classes of
separable $C^{*}$-algebras to be classified by the Elliott invariant are
friendly to $KK$-theory. So we must also assume that the $C^{*}$-algebras to
be classified by the Elliott invariant are in the UCT class.

##  3 Abstract approach

Finite $C^{*}$-algebras discussed in the previous section are presented as
inductive limits of some basic building blocks such as finite dimensional
$C^{*}$-algebras, or some (sub)-homogeneous $C^{*}$-algebras. However, some
naturally arising simple $C^{*}$-algebras, such as crossed product C*-algebras
$C(X)\rtimes_{\alpha}\mathbb{Z}$ for minimal homeomorphisms ${\alpha}:X\to X$,
do not have the appearance of inductive limits of some simpler forms.

######  Definition 3.1.

Let $A$ be a $C^{*}$-algebra. Let $a\in A_{+}.$ Denote by ${\rm Her}(a)$ the
hereditary $C^{*}$-subalgebra $\overline{aAa}.$ If $a,b\in A_{+},$ we write
$a\lesssim b$ ($a$ is Cuntz subequivalent to $b$), if there exists a sequence
of elements $x_{n}\in A$ such that $a=\lim_{n\to\infty}x_{n}^{*}x_{n}$ and
$x_{n}x_{n}^{*}\in{\rm Her}(b).$ If both $a\lesssim b$ and $b\lesssim a$ hold,
then we write $a\sim b$ and say $a$ is Cuntz equivalent to $b$. The Cuntz
equivalence class represented by $a$ will be denoted by $\langle a\rangle.$

The following definition was inspired by the papers [40], [24], [59], [61],
[22], [34], and [94].

######  Definition 3.2 (Lin, Definition 3.6.2 of [65], also [62] and [63]).

Let $A$ be a unital simple $C^{*}$-algebra. $A$ is said to have tracial rank
at most $k,$ if, for any $\varepsilon>0,$ any finite subset ${\cal F}\subset
A$ and any given $a\in A_{+}\setminus\\{0\\},$ there exists a
$C^{*}$-subalgebra $B\in{\cal I}_{k}$ with $p=1_{B}$ such that

(i) $\|px-xp\|<\varepsilon\,\,\,{\rm for\,\,\,all}\,\,\,x\in{\cal F},$

(ii) $pxp\in_{\varepsilon}B\,\,\,{\rm for\,\,\,all}\,\,\,x\in{\cal F},$ and

(iii) $p\lesssim a.$

If $A$ has tracial rank at most $k,$ we will write $TR(A)\leq k.$ If
$TR(A)\leq k$ but $TR(A)\not\leq k-1,$ we say that $A$ has tracial rank $k$
and write $TR(A)=k.$ In particular, if $k=0,$ we say that $A$ has tracial rank
zero and write $TR(A)=0.$

If $TR(A)\leq k$ for some $k\in\mathbb{N},$ then $A$ has stable rank one and
$K_{0}(A)$ is weakly unperforated. If $TR(A)=0,$ then $A$ also has real rank
zero (see [62] and [70]).

It was proved in [22] that a unital simple AH-algebra with slow dimension
growth and with real rank zero has tracial rank zero.

######  Theorem 3.3 (Lin [68]).

Let $A$ and $B$ be unital simple $C^{*}$-algebras of tracial rank zero in the
UCT class. Then $A\cong B$ if and only if

$\displaystyle(K_{0}(A),K_{0}(A)_{+},[1_{A}],K_{1}(A))\cong(K_{0}(B),K_{0}(B)_{+},[1_{B}],K_{1}(B)).$

It is proved in [66] that a unital simple AH-algebra $A$ which has real rank
zero, stable rank one, and weakly unperforated $K_{0}(A)$ must have tracial
rank zero. In particular, together with Theorem 2.6, all unital simple AH-
algebras with slow dimension growth and real rank zero have tracial rank zero
(and are therefore classifiable by Theorem 3.3 (compare Theorem 2.7)). It
should be mentioned that a unital separable simple exact $C^{*}$-algebra $A$
with real rank zero, stable rank one, and weakly unperforated $K_{0}(A)$ has
the property that $\rho_{A}(K_{0}(A))$ is dense in $\operatorname{Aff}(T(A))$
(see Lemma III 3.4 of [5]). Many other simple $C^{*}$-algebras have been
proved to have tracial rank zero, for example, in [112], W. Winter showed that
a unital separable simple $C^{*}$-algebra with finite decomposition rank and
real rank zero has tracial rank zero (see [84] for further examples). For
crossed product $C^{*}$-algebras, let us mention the next result.

######  Theorem 3.4 (Lin-Phillips, [80]).

Let $X$ be an infinite, compact finite dimensional metrizable space and
${\alpha}:X\to X$ be a minimal homeomorphism. Then the crossed product
$A=C(X)\rtimes_{\alpha}\mathbb{Z}$ has tracial rank zero if and only if
$\rho_{A}(K_{0}(A))$ is dense in $\operatorname{Aff}(T(A)).$

In particular, suppose that $X$ is a connected compact metric space with
finite dimension, and ${\alpha}:X\to X$ is minimal and uniquely ergodic; then
$A=C(X)\rtimes_{\alpha}\mathbb{Z}$ has tracial rank zero if the rotation
numbers of $(X,{\alpha})$ contain an irrational value ([80]). Consequently
every irrational rotation algebra has tracial rank zero. Thus, together with
Theorem 3.3, we recover the result of Evans-Elliott ([21]) that every
irrational rotation $C^{*}$-algebra is an inductive limit of circle algebras
which has real rank zero. Theorem 3.4 brought a large part of crossed product
$C^{*}$-algebras into the classifiable class of simple $C^{*}$-algebras.

Later, with Theorem 3.3, we move beyond the class of simple $C^{*}$-algebras
of real rank zero.

######  Theorem 3.5 (Lin [70]).

Let $A$ and $B$ be unital simple $C^{*}$-algebras of tracial rank at most one
in the UCT class. Then $A\cong B$ if and only if

$\displaystyle{\rm Ell}(A)\cong{\rm Ell}(B).$

It was proved in [34] that a unital simple AH-algebra with very slow dimension
growth has tracial rank at most one. The classification of C*-algebras which
can be tracially approximated by splitting interval algebras was also obtained
in [86], [87], [88], and [89].

Applying a remarkable work of Winter’s, his deformation result ([114]), as
pointed out, by Winter, led to the classification of ${\cal Z}$-stable
$C^{*}$-algebras $A$ with the property that $A\otimes U$ has finite tracial
rank for an arbitrary infinite dimensional UHF-algebra $U$.

Let us say, from now on, that $A$ has rational tracial rank $k,$ if $A\otimes
F$ has tracial rank $k$ for some infinite dimensional UHF-algebra $F.$

######  Theorem 3.6 (Lin-Niu [77]).

Let $A$ and $B$ be unital separable simple ${\cal Z}$-stable $C^{*}$-algebras
in the UCT class. Suppose that $A$ and $B$ has rational tracial rank zero.
Then $A\cong B$ if and only if

$\displaystyle{\rm Ell}(A)\cong{\rm Ell}(B).$ (e 3.1)

One of the important features of the theorem above is that the class of
$\mathcal{Z}$-stable unital separable simple $C^{*}$-algebras with rational
tracial rank zero includes the Jiang-Su algebra ${\cal Z}$ (i.e.
$TR(\mathcal{Z}\otimes F)=0$ if $F$ is UHF), and it is the only algebra in the
class with that invariant (the same as $\mathbb{C}$).

Toms and Winter proved the following generalization of Theorem 3.4 above.

######  Theorem 3.7 (Toms-Winter, [108]).

Let $X$ be an infinite, compact, finite dimensional metrizable space, and
${\alpha}:X\to X$ be a minimal homeomorphism and
$A=C(X)\rtimes_{\alpha}\mathbb{Z}.$ Then $A$ has rational tracial rank zero if
and only if its projections separate traces.

One may note that Theorem [108] above brought these crossed products into the
classifiable $C^{*}$-algebras, in particular, all crossed products
$C^{*}$-algebras of the form $C(X)\rtimes_{\alpha}\mathbb{Z}$ for which
${\alpha}$ is minumal and uniquely ergodic.

With further development of the so-called Basic Homotopy Lemma ([73]) and
asymptotic unitary equivalence of homomorphisms (see [71]), we reached the
following classification result.

######  Theorem 3.8 (Lin [74]).

Let $A$ and $B$ be unital separable amenable simple ${\cal Z}$-stable
$C^{*}$-algebras with rational tracial rank at most one in the UCT class. Then
$A\cong B$ if and only if

$\displaystyle{\rm Ell}(A)\cong{\rm Ell}(B).$ (e 3.2)

We also have the following range theorem.

######  Theorem 3.9 (Lin-Niu [78]).

Let $(G_{0},(G_{0})_{+},u)$ be a countable ordered weakly unperforated and
rationally Riesz group (see Proposition 5.8 and Definition 5.4 of [78]), let
$G_{1}$ be a countable abelian group, let $\Delta$ be a metrizable Choquet
simplex and let $\rho:G_{0}\to\operatorname{Aff}(\Delta)$ be an order
preserving homomorphism inducing a surjective affine continuous map
$\rho^{\sharp}:\Delta\to S(G_{0},u)$ such that
$\rho^{\sharp}(\partial_{e}(\Delta))=\partial_{e}(S(G,u)).$ Then there exists
one (and exactly one, up to isomorphism) unital simple amenable
$\mathcal{Z}$-stable C*-algebra $A$ with rational tracial rank at most one
such that

$\mathrm{Ell}(A)=((G_{0},(G_{0})_{+},u),G_{1},\Delta,\rho).$

This range theorem confirms that the class of rational tracial rank at most
one contains vast members of ${\cal Z}$-stable $C^{*}$-algebras.

Unfortunately, this range theorem does not cover all possible Elliott
invariants for unital separable simple ${\cal Z}$-stable $C^{*}$-algebras.
There are examples of unital separable simple ${\cal Z}$-stable
$C^{*}$-algebras $A$ for which $K_{0}(A)$ does not have the rational Riesz
property and the pairing $\rho^{\sharp}$ does not map extreme points to
extreme points.

Let us now mention another non-commuative dimension for $C^{*}$-algebras.

######  Definition 3.10.

Let $A$ and $B$ be C*-algebras. Recall ([52]) that a completely positive map
$\varphi:A\to B$ is said to have order zero if

$ab=0\ \Rightarrow\ \varphi(a)\varphi(b)=0,\quad a,b\in A.$

######  Definition 3.11 ([52], [115]).

A C*-algebra $A$ has nuclear dimension at most $n$, if there exists a net
$(F_{\lambda},\psi_{\lambda},\varphi_{\lambda})$, $\lambda\in\Lambda$, such
that the $F_{\lambda}$ are finite dimensional C*-algebras, and such that
$\psi_{\lambda}:A\to F_{\lambda}$ and $\varphi_{\lambda}:F_{\lambda}\to A$ are
completely positive maps satisfying
(1) $\varphi_{\lambda}\circ\psi_{\lambda}\to\operatorname{id}_{A}$ pointwise
(in norm),
(2) $\|\psi_{\lambda}\|\leq 1$, and
(3) for each $\lambda$, there is a decomposition
$F_{\lambda}=F_{\lambda}^{(0)}\oplus\cdots\oplus F_{\lambda}^{(n)}$ such that
each restriction $\varphi_{\lambda}|_{F_{\lambda}^{(j)}}$ is a contractive
order zero map.

Moreover, if the the map $\varphi_{\lambda}$ can be chosen to be contractive
itself, then $A$ is said to have decomposition rank at most $n$.

######  Theorem 3.12 (Winter [113]).

Let A be a separable, simple, nonelementary, unital $C^{*}$-algebra with
finite nuclear dimension. Then $A$ is ${\cal Z}$-stable.

We will present a classification theorem for separable amenable simple ${\cal
Z}$-stable $C^{*}$-algebras.

##  4 The end of the beginning, unital case

Recall that the Jiang-Su algebra ${\cal Z}$ ([50] and see Theorem 2.12 above)
constructed in 1998 during the development of the Elliott program is an
infinite dimensional unital amenable $C^{*}$-algebra in the UCT class with the
feature that its Elliott invariant is exactly the same as that of the complex
field. As stated earlier (see [39] and Theorem 2.13 above), for any separable
simple $C^{*}$-algebra $A,$ $A\otimes{\cal Z}$ and $A$ have the same tracial
structure and $K_{i}(A)=K_{i}(A\otimes{\cal Z})$ ($i=0,1$). Moreover,
$A\otimes{\cal Z}$ and $A$ are in the same one of the three cases mentioned at
the end of Section 1. Since $A\otimes{\cal Z}$ is ${\cal Z}$-stable, i.e.,
$A\otimes{\cal Z}\cong(A\otimes{\cal Z})\otimes{\cal Z},$ naturally one
studies simple ${\cal Z}$-stable $C^{*}$-algebras (see also [107] for further
non-${\cal Z}$-stable examples). Fortunately, a separable nuclear simple
$C^{*}$-algebra is ${\cal Z}$-stable if and only if it has finite nuclear
dimension (see [113], [81], [82], [103], [7], [105], and [8]).

######  Definition 4.1 (See [31] and [20]).

Let $F_{1}$ and $F_{2}$ be two finite dimensional $C^{*}$-algebras. Suppose
that there are two unital homomorphisms $\varphi_{0},\varphi_{1}:F_{1}\to
F_{2}.$ Consider the mapping torus $M_{\varphi_{0},\varphi_{1}}:$

$A=A(F_{1},F_{2},\varphi_{0},\varphi_{1})=\\{(f,g)\in C([0,1],F_{2})\oplus
F_{1}:f(0)=\varphi_{0}(g)\,\,\,{\rm and}\,\,\,f(1)=\varphi_{1}(g)\\}.$

These $C^{*}$-algebras were introduced into the Elliott program by Elliott and
Thomsen ([31]), and in [20], Elliott used this class of $C^{*}$-algebras and
some other building blocks with 2-dimensional spectra to realize any weakly
unperforated simple ordered group with order unit as the $K_{0}$-group of a
simple ASH $C^{*}$-algebra. Denote by ${\cal C}$ the class of all unital
$C^{*}$-algebras of the form $A=A(F_{1},F_{2},\varphi_{0},\varphi_{1})$ (which
includes all finite dimensional $C^{*}$-algebras). These $C^{*}$-algebras will
be called Elliott-Thomsen building blocks as well as 1-dimensional
noncommutative CW complexes.

Let $\lambda:A\to C([0,1],F_{2})$ be defined by $\lambda((f,a))=f.$ There is a
canonical map $\pi_{e}:A\to F_{1}$ defined by $\pi_{e}(f,g)=g$ for all pair
$(f,g)\in A.$ It is a surjective map.

If $A\in{\cal C}$, then $A$ is the pull-back corresponding to the diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\scriptstyle{\pi_{e}}$$\textstyle{C([0,1],F_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{0},\pi_{1})}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\varphi_{0},\varphi_{1})}$$\textstyle{F_{2}\oplus
F_{2}\,.}$ (e 4.1)

######  Definition 4.2.

Let $A$ be a unital separable simple $C^{*}$-algebra. We say $A$ has
generalized tracial rank at most one, if, for any $\varepsilon>0,$ any finite
subset ${\cal F}\subset A$ and any $a\in A_{+}\setminus\\{0\\},$ there is a
$C^{*}$-subalgebra $B\in{\cal C}$ such that $p=1_{B}$ and

(i) $\|px-xp\|<\varepsilon\,\,\,{\rm for\,\,\,all}\,\,\,x\in{\cal F},$

(ii) $pxp\in_{\varepsilon}B\,\,\,{\rm for\,\,\,all}\,\,\,x\in{\cal F}$, and

(iii) $p\lesssim a.$

If $A$ has generalized tracial rank at most one, we may write $gTR(A)\leq 1.$
If there exists an infinite dimensional UHF-algebra $F$ such that
$gTR(A\otimes F)\leq 1,$ we say $A$ has rationally generalized tracial rank at
most one.

The introduction of the class of unital separable amenable simple
$C^{*}$-algebras of rational generalized tracial rank at most one is
important. From the following range theorem, we conclude that this class of
simple $C^{*}$-algebras exhausts all possible Elliott invariants for unital
separable simple ${\cal Z}$-stable $C^{*}$-algebras.

######  Theorem 4.3 (Gong-Lin-Niu Theorem 13.50 of [45]).

For any simple weakly unperforated Elliott invariant
$\big{(}(G,G_{+},u),~{}K,\Delta,r\big{)}$, where $(G,G_{+},u)$ is a weakly
unperforated countable simple ordered group $G$ with positive cone $G_{+}$ and
with a non-zero element $u,$ $K$ is any countable abelian group, $\Delta$ any
compact metrizable Choquet simplex, and $r:G\to\operatorname{Aff}(\Delta)$ any
order preserving homomorphism (with the strict order on
$\operatorname{Aff}(\Delta)$), there is a unital simple ${\cal Z}$-stable
$C^{*}$-algebra $A$ with rational tracial rank at most one which is also an
$ASH$-algebra such that

$\big{(}(K_{0}(A),K_{0}(A)_{+},[1_{A}]),~{}K_{1}(A),T(A),\rho_{A}\big{)}\cong\big{(}(G,G_{+},u),~{}K,\Delta,r\big{)}.$

######  Remark 4.4.

In the case of finite separable simple unital ${\cal Z}$-stable
$C^{*}$-algebras, $K_{0}(A)_{+}$ is determined by $[1_{A}]$ and the pairing
$\rho_{A}:K_{0}(A)\to\operatorname{Aff}(T(A).$ Therefore the invariant could
be simplified as

$\displaystyle(K_{0}(A),[1_{A}],K_{1}(A),T(A),\rho_{A}).$ (e 4.2)

However, we keep the positive cone $K_{0}(A)_{+}$ following the early
tradition.

Let us present the following isomorphism theorem:

######  Theorem 4.5 (Gong-Lin-Niu [46]).

Let $A$ and $B$ be unital separable amenable simple ${\cal Z}$-stable
$C^{*}$-algebras with rational generalized tracial rank at most one in the UCT
class. Then $A\cong B$ if and only if

$\displaystyle{\rm Ell}(A)\cong{\rm Ell}(B).$ (e 4.3)

The question is then whether every finite unital separable amenable simple
${\cal Z}$-stable $C^{*}$-algebra has rational generalized tracial rank at
most one.

######  Theorem 4.6 (Castillejos, Evington, Tikuisis, White, and Winter [7]).

Let A be an infinite dimensional, separable, simple, unital, nuclear
$C^{*}$-algebra. Then the following statements are equivalent:

(i) $A$ has finite nuclear dimension;

(ii) $A$ is ${\cal Z}$-stable, i.e., $A\cong A\otimes{\cal Z}.$

Let us state the following theorem.

######  Theorem 4.7 (Elliott-Gong-Lin-Niu Theorem 4.9 of [25]).

Let $A$ be a unital simple separable amenable (non-zero) C*-algebra which
satisfies the UCT. Then the following properties are equivalent:

(1) ${\mathrm{gTR}}(A\otimes Q)\leq 1;$

(2) $A\otimes Q$ has finite nuclear dimension and ${\rm T}(A\otimes Q)={\rm
T}_{{\rm qd}}(A\otimes Q)\not=\O;$

(3) the decomposition rank of $A\otimes Q$ is finite.

(Here $Q$ is the universal UHF algebra and ${\rm T}_{{\rm qd}}(A\otimes Q)$ is
the set of quasidiagonal traces.)

It follows from a result of Tikuisis, White, and Winter [106] that every
tracial state on a unital simple separable amenable $C^{*}$-algebra which
satisfies the UCT is quasidiagonal. Since $A\otimes Q$ is always ${\cal
Z}$-stable, by Theorem 4.6, $A\otimes Q$ has finite nuclear dimension.
Therefore (2) holds for any finite unital separable amenable simple
$C^{*}$-algebra $A$ in the UCT class, whence (1) in Theorem 4.7 always holds.
From this we obtain the following theorem.

######  Theorem 4.8 (Elliott-Gong-Lin-Niu [25]).

Let $A$ be a unital simple separable amenable $C^{*}$-algebra which satisfies
the UCT. Then $A$ has rational generalized tracial rank at most one.

Together with the isomorphism theorem 4.5, as well as the classification of
purely infinite simple $C^{*}$-algebras of Kirchberg and Phillips [51] and
[93]), we further obtain the following:

######  Corollary 4.9 (Elliott-Gong-Lin-Niu [25]).

Let $A$ and $B$ be unital separable simple amenable ${\cal Z}$-stable
$C^{*}$-algebras in the UCT class. Then $A\cong B$ if and only if

$\displaystyle{\rm Ell}(A)\cong{\rm Ell}(B).$ (e 4.4)

##  5 The end of the beginning, non-unital case

Next we will discuss the non-unital case. We will mainly discuss the stably
projectionless case. In fact, the classification of non-unital separable
amenable simple ${\cal Z}$-stable $C^{*}$-algebras in the UCT class which are
not stably projectionless can easily be reduced to the unital case (see the
proof of Theorem 5.5 below).

However, there are separable simple $C^{*}$-algebras $A$ which are inductive
limits of sub-homogeneous $C^{*}$-algebras such that $A\otimes{\cal K}$ admits
no non-zero projections. One of them, $W,$ a Razak algebra, has a unique
tracial state and has zero $K_{0}$ and zero $K_{1}$ ([95]). In other words,
Case 2 mentioned in Section 1 does occur.

The study of stably projectionless simple $C^{*}$-algebras actually has a long
history (see Kishimoto’s 1980 paper [53], and Blackadar and Cuntz’s 1982 paper
[3], for example). Stably projectionless simple $C^{*}$-algebras can naturally
occur, for example, in the study of flow actions (see [53], [54], [96], and
[105]). Tsang ([109]) showed that any metrizable Choquet simplex can be the
tracial state space for some stably projectionless simple amenable
$C^{*}$-algebra. In fact, in [42], we show that there is a unique separable
stably projectionless simple $C^{*}$-algebra ${\cal Z}_{0}$ with finite
nuclear dimension in the UCT class and with $K_{0}({\cal Z}_{0})=\mathbb{Z},$
$K_{1}({\cal Z}_{0})=\\{0\\}$, and with a unique tracial state (such
$C^{*}$-algebras were known to exist). It turns out that, for any separable
simple nuclear $C^{*}$-algebra $A,$ $A\otimes{\cal Z}_{0}$ is stably
projectionless. Furthermore, for any abelian group $G_{0},$ any compact
metrizable Choquet simplex $\Delta$, and any homomorphism
$\rho:G_{0}\to\operatorname{Aff}(\Delta),$ the space of continuous real affine
functions on $\Delta$, such that
$\rho(G_{0})\cap\operatorname{Aff}_{+}(\Delta)=\\{0\\},$ there exists a
separable stably projectionless simple nuclear ${\cal Z}$-stable
$C^{*}$-algebra $A$ such that $(K_{0}(A),T(A),\rho_{A})=(G_{0},\Delta,\rho)$
(and with arbitrarily given $K_{1}(A)$) (see [20] and [43]).

Since the stably projectionless case is not included in [45] and [46], we will
provide a few more details.

###  5.1 The Elliott invariant and its range

We start with the description of the invariant.

######  Definition 5.1.

Let $A$ be a $C^{*}$-algebra. Denote by $T(A)$ the tracial state space of $A$
(which could be an empty set). Denote by $T_{f}(A)$ the set of faithful traces
of $A$ (if $a\in A_{+}\setminus\\{0\\}$ and $\tau\in T_{f}(A),$ then
$\tau({{a}})>0$). Let ${\tilde{T}}(A)$ denote the cone of densely defined,
positive lower semi-continuous traces on $A$ equipped with the topology of
point-wise convergence on elements of the Pedersen ideal ${\rm Ped}(A)$—the
minimal dense hereditary ideal of $A.$ Let $B$ be another $C^{*}$-algebra with
$\tilde{T}(B)\not=\\{0\\}$ and let $\varphi:A\to B$ be a homomorphism.

In what follows we will also write $\varphi$ for
$\varphi\otimes\operatorname{id}_{M_{k}}:M_{k}(A)\to M_{k}(B)$ whenever it is
convenient. We will write $\varphi_{T}:\tilde{T}(B)\to\tilde{T}(A)$ for the
induced continuous affine map. Denote by $\tilde{T}^{b}(A)$ the subset of
$\tilde{T}(A)$ of traces which are bounded on $A.$ Of course
$T(A)\subset\tilde{T}^{b}(A).$ Set
$T_{0}(A):=\\{\tau\in\tilde{T}(A):\|\tau\|\leq 1\\}.$ It is a compact convex
subset of $\tilde{T}(A).$

Let $r\geq 1$ be an integer and $\tau\in{\tilde{T}}(A).$ We will continue to
write $\tau$ on $A\otimes M_{r}$ for $\tau\otimes{\rm Tr},$ where ${\rm Tr}$
is the standard trace on $M_{r}.$ Let $S$ be a convex subset (of a convex
topological cone $C$ with a Choquet simplex as a base). We assume that the
convex cone contains 0. Denote by $\operatorname{Aff}(C)$ the set of affine
real continuous functions on $C$ which vanish at $0.$ Define (see [96])

$\displaystyle\operatorname{Aff}(S)$ $\displaystyle=$
$\displaystyle\\{f|_{S}:f\in\operatorname{Aff}(C)\\}\,\,\,{\rm
and}\,\,\,\operatorname{Aff}(S)_{+}=\\{f\in\operatorname{Aff}(S):f\geq 0\\},$
(e 5.1) $\displaystyle\operatorname{Aff}_{+}(S)$ $\displaystyle=$
$\displaystyle\\{f\in\operatorname{Aff}(S):f(\tau)>0\,\,{\rm for}\,\,\tau\in
S\setminus\\{0\\}\\}\cup\\{0\\},$ (e 5.2) $\displaystyle{\rm LAff}(S)_{+}$
$\displaystyle=$
$\displaystyle\\{f:S\to[0,\infty]:\exists\\{f_{n}\\},f_{n}\nearrow
f,\,\,f_{n}\in\operatorname{Aff}(S)_{+}\\},$ (e 5.3) $\displaystyle{\rm
LAff}_{+}(S)$ $\displaystyle=$
$\displaystyle\\{f:S\to[0,\infty]:\exists\\{f_{n}\\},f_{n}\nearrow
f,\,\,f_{n}\in\operatorname{Aff}_{+}(S)\\}.$ (e 5.4)

To include the case that $S=\\{0\\},$ we view that it is the only point in the
zero cone. In that case, we define
$\operatorname{Aff}(S)=\operatorname{Aff}_{+}(S)=\\{0\\}={\rm LAff}_{+}(S).$
For the greater part of this paper, $S={\widetilde{T}}(A)$ or $S=T(A),$ or
$S=T_{0}(A)$ in the above definition will be used. Recall that
$0\in{\tilde{T}}(A)$ and if ${{g}}\in{\rm LAff}(\tilde{T}(A)),$ then
${{g}}(0)=0.$

######  Definition 5.2.

For any $\varepsilon>0,$ define $f_{\varepsilon}\in{{C([0,\infty))_{+}}}$ by
$f_{\varepsilon}(t)=0$ if $t\in[0,\varepsilon/2],$ $f_{\varepsilon}(t)=1$ if
$t\in[\varepsilon,\infty)$, and $f_{\varepsilon}(t)$ is linear in
$(\varepsilon/2,\varepsilon).$

Let $A$ be a $C^{*}$-algebra and $\tau$ a trace on $A$. For each $a\in A_{+},$
define $d_{\tau}(a)=\lim_{\varepsilon\to 0}{{\tau(f_{\varepsilon}(a)).}}$ Note
that $f_{\varepsilon}(a)\in{\rm Ped}(A)$ for all $a\in A_{+}$ (see Theorem
5.6.1 of [91] and its proof).

Let $S$ be a convex subset of $\widetilde{T}(A)$ and $a\in M_{n}(A)_{+}.$ The
function $\hat{a}(s)=s(a)$ (for $s\in S$) is an affine function. Define
$\widehat{\langle a\rangle}(s)=d_{s}(a)=\lim_{\varepsilon\to
0}s(f_{\varepsilon}(a))$ (for $s\in S$), which is a lower semicontinuous
function. If $a\in{\rm Ped}(A)_{+},$ then $\hat{a}$ is in
$\operatorname{Aff}(S)_{+}$ and $\widehat{\langle a\rangle}\in{\rm
LAff}(S)_{+}$ in general (see 5.1). If $A$ is simple, then $\widehat{\langle
a\rangle}\in{\rm LAff}_{+}(S).$ Note that $\hat{a}$ is different from
$\widehat{\langle a\rangle}.$ In most cases, $S$ is ${\tilde{T}}(A),$
$T_{0}(A),$ or $T(A).$ Note also that, when $A$ is simple, there is a natural
map from ${\rm Cu}(A)$ to ${\rm LAff}_{+}(\tilde{T}(A))$, sending $\langle
a\rangle$ to $\widehat{\langle a\rangle}$.

######  5.3.

If $A$ is a unital $C^{*}$-algebra and $T(A)\not=\emptyset,$ then there is a
canonical homomorphism $\rho_{A}:K_{0}(A)\to\operatorname{Aff}(T(A)).$ Now
consider the case that $A$ is not unital. Let
$\pi_{\mathbb{C}}^{A}:\tilde{A}\to\mathbb{C}$ denote the quotient map. Suppose
that $T(A)\not=\emptyset.$ Let
$\tau_{\mathbb{C}}:=\tau_{\mathbb{C}}^{A}:\tilde{A}\to\mathbb{C}$ denote the
tracial state which factors through $\pi_{\mathbb{C}}^{A}.$ Then

$\displaystyle
T(\tilde{A})=\big{\\{}t{{\tau_{{}_{\mathbb{C}}}^{A}}}+(1-t)\tau:~{}t\in[0,1],~{}\tau\in{{T(A)}}\big{\\}}.$
(e 5.5)

The map ${{T(A)}}\hookrightarrow{{T(\tilde{A})}}$ induces a map
${{\operatorname{Aff}(T(\tilde{A}))}}\to\operatorname{Aff}(T(A))$. Then the
map $\rho_{\tilde{A}}:K_{0}({{\tilde{A}}})\to\operatorname{Aff}(T(\tilde{A}))$
induces a homomorphism $\rho^{\prime}:~{}K_{0}(A)\to\operatorname{Aff}(T(A))$
by

$\displaystyle{{\rho^{\prime}:~{}K_{0}(A)\to
K_{0}({{\tilde{A}}})\stackrel{{\scriptstyle\rho_{\tilde{A}}}}{{\longrightarrow}}\operatorname{Aff}(T(\tilde{A}))\to\operatorname{Aff}(T(A)).}}$
(e 5.6)

However, in the case that $A\not={\rm Ped}(A),$ we will not use
$\rho^{\prime}$ in general, as it is possible that $T(A)=\emptyset$ but
${\tilde{T}}(A)$ is rich (consider the case $A\cong A\otimes{\cal K}$).

######  Definition 5.4.

[Definition 2.15 of [43]] Let $A$ be a $C^{*}$-algebra with
$\tilde{T}(A)\not=\\{0\\}.$ If $\tau\in\tilde{T}(A)$ is bounded on $A,$ then
$\tau$ can be extended naturally to a trace on $\tilde{A}.$ Recall that
$\tilde{T}^{b}(A)$ is the set of bounded traces on $A.$ Denote by
$\rho_{A}^{b}:K_{0}(A)\to\operatorname{Aff}(\tilde{T}^{b}(A))$ the
homomorphism defined by $\rho_{A}^{b}([p]-[q])=\tau(p)-\tau(q)$ for all
$\tau\in\tilde{T}^{b}(A)$ and for projections $p,q\in M_{n}(\tilde{A})$ (for
some integer $n\geq 1$) and $\pi_{\mathbb{C}}^{A}(p)=\pi_{\mathbb{C}}^{A}(q).$
Note $p-q\in M_{n}(A).$ Therefore $\rho_{A}^{b}([p]-[q])$ is continuous on
$\tilde{T}^{b}(A).$ In the case that $\tilde{T}^{b}(A)=\tilde{T}(A),$ for
example, $A={\rm Ped}(A),$ we write $\rho_{A}:=\rho_{A}^{b}.$

Let $A$ be a $\sigma$-unital $C^{*}$-algebra with a strictly positive element
$0\leq e\leq 1.$ Put $e_{n}:=f_{1/2^{n}}(e).$ Then $\\{e_{n}\\}$ forms an
approximate identity for $A.$ Note $e_{n}\in{\rm Ped}(A)$ for all $n.$ Set
$A_{n}={\rm Her}(e_{n}):=\overline{e_{n}Ae_{n}}.$ Denote by
$\iota_{n}:A_{n}\to A_{n+1}$ and $j_{n}:A_{n}\to A$ the embeddings. These
extend to $\iota_{n}^{\sim}:\tilde{A}_{n}\to\tilde{A}_{n+1}$ and
$j_{n}^{\sim}:\tilde{A}_{n}\to\tilde{A}$ unitally. Note that $e_{n}\in{\rm
Ped}(A_{n+1}).$ Thus $\iota_{n}$ and $j_{n}$ induce continuous cone maps
${\iota_{n}}_{T}^{b}:\tilde{T}^{b}(A_{n+1})\to\tilde{T}^{b}(A_{n})$ and
${j_{n}}_{T}:\tilde{T}(A)\to\tilde{T}^{b}(A_{n})$ (defined by
${\iota_{n}}_{T}^{b}(\tau)(a)=\tau(\iota_{n}(a))$ for
$\tau\in{{\tilde{T}^{b}}}(A_{n+1}),$ and ${j_{n}}_{T}(\tau)(a)=\tau(j_{n}(a))$
for all $\tau\in\tilde{T}(A)$ and all $a\in A_{n}$), respectively. Denote by
$\iota_{n}^{\sharp}:\operatorname{Aff}(\tilde{T}^{b}(A_{n}))\to{{\operatorname{Aff}(\tilde{T}^{b}(A_{n+1}))}}$
and
$j_{n}^{\sharp}:\operatorname{Aff}(\tilde{T}^{b}(A_{n}))\to\operatorname{Aff}(\tilde{T}(A))$
the induced continuous linear maps. Recall that $\bigcup_{n=1}^{\infty}A_{n}$
is dense in ${\rm Ped}(A).$ A direct computation shows that one may obtain the
following inverse direct limit of convex topological cones (with continuous
cone maps):

$\displaystyle\tilde{T}^{b}(A_{1})\stackrel{{\scriptstyle{\iota_{1}}_{T}^{b}}}{{\longleftarrow}}\tilde{T}^{b}(A_{2})\stackrel{{\scriptstyle{\iota_{2}}_{T}^{b}}}{{\longleftarrow}}\tilde{T}^{b}(A_{3})\cdots\longleftarrow\cdots\longleftarrow\tilde{T}(A),$
(e 5.7)

which induces the following commutative diagram:

$\displaystyle\operatorname{Aff}(\tilde{T}^{b}(A_{1}))\stackrel{{\scriptstyle\iota_{1}^{\sharp}}}{{\longrightarrow}}\operatorname{Aff}(\tilde{T}^{b}(A_{2}))\stackrel{{\scriptstyle{\iota_{2}}^{\sharp}}}{{\longrightarrow}}\operatorname{Aff}(\tilde{T}^{b}(A_{3}))\cdots\longrightarrow\cdots\longrightarrow\operatorname{Aff}(\tilde{T}(A)).$
(e 5.8)

Hence one also has the following commutative diagram:

$\textstyle{K_{0}(A_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{A_{1}}}$$\scriptstyle{\iota_{1*o}}$$\textstyle{K_{0}(A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2*o}}$$\scriptstyle{\rho_{A_{2}}}$$\textstyle{K_{0}(A_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{A_{3}}}$$\textstyle{\cdots
K_{0}({{A}})}$$\textstyle{\operatorname{Aff}(\tilde{T}^{b}({{A}}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\sharp}_{1,2}}$$\textstyle{\operatorname{Aff}(\tilde{T}^{b}({{A}}_{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}^{\sharp}}$$\textstyle{\operatorname{Aff}(\tilde{T}^{b}({{A}}_{3}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\operatorname{Aff}(\tilde{T}({{A}})).}$

Thus one obtains a homomorphism
$\rho:K_{0}(A)\to\operatorname{Aff}(\tilde{T}(A)).$ It should be noted that,
when $A$ is simple, $\tilde{T}^{b}(A_{n})=\tilde{T}(A_{n})$ for all $n$ (see
Definition 2.15 of [43] for more details). Moreover, the map $\rho$ does not
depend on the choice of $\\{e_{n}\\}.$ We will write $\rho_{A}:=\rho.$ In the
case that $T(A)$ generates ${\tilde{T}}(A)$ such as the case that $A$ has
continuous scale, we may also write
$\rho_{A}:K_{0}(A)\to\operatorname{Aff}(T(A))$ by restricting $\rho_{A}(x)$ to
$T(A)\subset{\tilde{T}}(A)$ and for all $x\in K_{0}(A).$

Now let us present an isomorphism theorem for non-unital separable amenable
simple ${\cal Z}$-stable $C^{*}$-algebras in the UCT class which are not
stably projectionless. Suppose that $A$ has a strictly positive element
$e_{A}.$ Denote by $\widetilde{\langle e_{A}\rangle}$ the dimension function
$\widetilde{\langle e_{A}\rangle}(\tau)=d_{\tau}(\langle e_{A}\rangle)$
($\tau\in\widetilde{T}(A)$). Note that the dimension function
$\widetilde{\langle e_{A}\rangle}$ is independent of the choice of the
strictly positive element $e_{A}.$

######  Theorem 5.5.

Let $A$ and $B$ be non-unital separable simple ${\cal Z}$-stable
$C^{*}$-algebras in the UCT class with strictly positive elements $e_{A}$ and
$e_{B},$ respectively. Suppose that $A\otimes{\cal K}$ contains a non-zero
projection. Then $A\cong B$ if and only if

$\displaystyle(K_{0}(A),\widetilde{\langle
e_{A}\rangle},\widetilde{T}(A),\rho_{A},K_{1}(A))\cong(K_{0}(B),\widetilde{\langle
e_{B}\rangle},\widetilde{T}(B),\rho_{B},K_{1}(B)).$ (e 5.9)

######  Definition 5.6.

Note that (e 5.9) means that there is an a group isomorphism
$\kappa_{0}:K_{0}(A)\to K_{0}(B),$ an affine homeomorphism
$\kappa_{T}:\widetilde{T}(A)\to\widetilde{T}(B)$ such that

$\displaystyle\rho_{B}(\kappa_{0}(x))(\tau)=\rho_{A}(x)(\kappa_{T}^{-1}(\tau))\,\,\,{\rm
for\,\,\,all}\,\,\,x\in K_{0}(A)\,\,\,{\rm and}\,\,\,\tau\in\widetilde{T}(B),$
(e 5.10)

and $\widetilde{\langle
e_{A}\rangle}(\kappa_{T}^{-1}(\tau))=\widetilde{\langle e_{B}\rangle}(\tau)$
for all $\tau\in\widetilde{T}(B),$ and a group isomorphism
$\kappa_{1}:K_{1}(A)\to K_{1}(B).$

Proof of Theorem 5.5:  Let

$\displaystyle\Gamma:(K_{0}(A),\widetilde{\langle
e_{A}\rangle},\widetilde{T}(A),\rho_{A},K_{1}(A))\cong(K_{0}(B),\widetilde{\langle
e_{B}\rangle},\widetilde{T}(B),\rho_{B},K_{1}(B))$

be an isomorphism.

Let $p\in A\otimes{\cal K}$ be a non-zero projection and set
$A_{1}=p(A\otimes{\cal K})p.$ Then $A_{1}$ is a unital simple separable
amenable ${\cal Z}$-stable $C^{*}$-algebra in the UCT class. Let $q\in
B\otimes{\cal K}$ be a projection such that $[q]=\kappa_{0}([p]).$ Put
$B_{1}=q(B\otimes{\cal K})q.$ By applying Theorem 4.9, there is an isomorphism
$\varphi:A_{1}\to B_{1}$ such that $\varphi$ induces $\Gamma.$ Note that $A$
is non-unital, and since ${\rm Ell}(B)\cong{\rm Ell}(A),$ $\Sigma(K_{0}(B))$
has either no maximum element or the maximum element $u\in\Sigma(K_{0}(B))$
does not have the property $\rho_{B}(u)=\widetilde{\langle e_{B}\rangle}.$
Hence $B$ is not unital. Consequently neither $\langle e_{A}\rangle$ nor
$\langle e_{B}\rangle$ can be represented by projections. We may view
$e_{A}\in A_{1}\otimes{\cal K}$ and $e_{B}\in B_{1}\otimes{\cal K}$ with
$\overline{e_{A}(A_{1}\otimes{\cal K})e_{A}}=A$ and
$\overline{e_{B}(B_{1}\otimes{\cal K})e_{B}}=B.$ On the other hand, we must
have $A\cong\overline{\psi(e_{A})(B_{1}\otimes{\cal K})\psi(e_{A})}$ and
$\widetilde{\langle\psi(e_{A})\rangle}=\widetilde{\langle e_{B}\rangle},$ and
both $\psi(e_{A})$ and $\langle e_{B}\rangle$ are not represented by
projections. Then, by Theorem 6.6 of [29], $\langle\psi(e_{A})\rangle=\langle
e_{B}\rangle.$ It follows from Proposition 3.3 of [97] (see also Theorem 3 of
[9] as well as Proposition 4.3 of [90]) that there exists a partial isometry
$v\in(B_{1}\otimes{\cal K})^{**}$ such that
$v\overline{\psi(e_{A})(B_{1}\otimes{\cal K})}\in B_{1}\otimes{\cal K},$
$v^{*}\overline{e_{B}(B_{1}\otimes{\cal K})e_{B}}\in B_{1}\otimes{\cal K},$
and $v^{*}v=p,$ the open projection corresponding to $\psi(e_{A})$, and
$vv^{*}=q,$ the open projection corresponding to $e_{B}.$ Then
$\varphi_{1}:A\to B$ defined by $\varphi_{1}(a)=v\psi(a)v^{*}$ for all $a\in
A$ is an isomorphism from $A$ to $B$ which carries $\Gamma.$

With Corollary 4.9 and Theorem 5.5, we complete the classification of Case
(1). The world is not perfect. As mentioned earlier, Case 2 does occur. Let us
now discuss the Case 2, the stably projectionless case.

######  5.7.

Recall from Definition 2.7 of [43] (see (2.8) of [43] also), that a (simple)
ordered group pairing is a triple $(G,T,\rho),$ where $G$ is a countable
abelian group, $T$ is a convex topological cone with a Choquet simplex as
base, and $\rho:G\to\operatorname{Aff}(T)$ is a homomorphism. Let us also add
the case that $T=\\{0\\}$ with $\rho=0.$ Define

$\displaystyle\operatorname{Aff}_{+}(T)^{+}=\\{f\in\operatorname{Aff}_{+}(T):f(t)>0,\,\,\,{\rm
whenever}\,\,\,t\not=0\\}.$ (e 5.11)

Note that, in the case that $T=\\{0\\},$
$\operatorname{Aff}_{+}(T)^{+}=\operatorname{Aff}_{+}(T)=\\{0\\}.$

Define $G^{+}=\\{g\in
G:\rho(g)\in\operatorname{Aff}_{+}(T)^{+}\\}\cup\\{0\\}.$ When
$T\not=\\{0\\},$ if $G^{+}\not=\\{0\\},$ then $(G,G^{+})$ is an ordered group
in the sense that $G^{+}\cap(-G^{+})=\\{0\\}$ and $G^{+}-G^{+}=G$. It has the
property that if $ng>0$ for some integer $n>0,$ then $g>0.$ In other words,
$(G,G^{+})$ is weakly unperforated. When $T=\\{0\\},$ recall that
$\operatorname{Aff}_{+}(T)^{+}=\\{0\\},$ and hence $G^{+}=G.$

The ordered group pairing mentioned above satisfies the following condition:
when $G\not=G^{+},$ either $G^{+}=\\{0\\},$ or $(G,G^{+})$ is a simple ordered
group, i.e., every element $g\in G^{+}\setminus\\{0\\}$ is an order unit.

A simple ordered group pairing with a distinguished set, or a scaled simple
ordered group pairing  is a quintuple $(G,\Sigma_{\rho}(G),T,s,\rho)$ such
that $(G,T,\rho)$ is a simple ordered group pairing, $s\in{\rm
LAff}_{+}(T)\setminus\\{0\\}$, or, $s=0,$ when $T=\\{0\\},$ and

$\displaystyle\Sigma_{\rho}(G):=\\{g\in G^{+}:\rho(g)<s\\}\,\,\,{\rm
or}\,\,\,\Sigma_{\rho}(G):=\\{g\in G^{+}:\rho(g)<s\\}\cup\\{u\\},$ (e 5.12)

where $\rho(u)=s$ and $\rho(g)<s$ means
$s-\rho(g)\in\operatorname{Aff}_{+}(T)$ and $s-\rho(g)\not=0.$ Here
$\Sigma_{\rho}(G)$ is the distinguished set. It could contain only one element
or even be an empty set. We also allow $\Sigma_{\rho}(G)=\\{0\\}.$ In the case
that $T=\\{0\\},$ $s=0.$ No element $g\in G^{+}$ with $\rho(g)<s.$ Hence when
$T=\\{0\\},$ if $\Sigma_{\rho}(G)=\\{g\in G^{+}:\rho(g)<s\\},$ then
$\Sigma_{\rho}(G)=\emptyset,$ and if $\Sigma_{\rho}(G)=\\{g\in
G^{+}:\rho(g)<s\\}\cup u$ with $\rho(u)=s,$ then $\Sigma_{\rho}(g)=\\{u\\}.$
Therefore, when $T=\\{0\\},$ there are two cases, either the distinguished set
$\Sigma_{\rho}(G)=\emptyset,$ or $\Sigma_{\rho}(G)=\\{u\\},$ a single element
(which could be zero).

Note that $s(\tau)$ could be infinite for some $\tau\in T.$ It is called a
unital scaled simple ordered group pairing, if $\Sigma_{\rho}(G)=\\{g\in
G^{+}:\rho(g)<s\\}\cup\\{u\\}$ with $\rho(u)=s,$ in which case, $u$ is called
the unit for $G.$ Note also that, in this case, $u$ is the maximum element of
$\Sigma_{\rho}(G),$ one may write $(G,u,T,\rho)$ for
$(G,\Sigma_{\rho}(G),T,s,\rho).$ On the other hand, $\Sigma_{\rho}(G)$ is
determined by $s$ if it has no unit. One may then write $(G,T,s,\rho)$ for
$(G,\Sigma_{\rho}(G),T,s,\rho)$ (see Theorem 5.2 of [43].)

Let $(G_{i},\Sigma_{\rho}(G_{i}),T_{i},s_{i},\rho_{i}),$ $i=1,2,$ be scaled
simple ordered group pairings. A map

$\Gamma_{0}:(G_{1},\Sigma_{\rho}(G_{1}),T_{1},s_{1},\rho_{1})\to(G_{2},\Sigma_{\rho}(G_{2}),T_{2},s_{2},\rho_{2})$

is said to be an isomorphism, if it consists of a group isomorphism
$\kappa_{0}:G_{1}\to G_{2}$ and an affine cone isomorphism
$\kappa_{T}:T_{1}\to T_{2}$ such that

$\displaystyle{{\rho_{2}}}(\kappa_{0}(g))(t)={{\rho_{1}(g)}}(\kappa_{T}^{-1}(t))\,\,\,{\rm
for\,\,\,all}\,\,\,g\in G_{1}\,\,\,{\rm and}\,\,\,t\in T_{2},\,\,\,{\rm
and}\,\,\,$ (e 5.13)
$\displaystyle\kappa_{0}(\Sigma_{\rho}(G_{1}))=\Sigma_{\rho}(G_{2})),\,\,\,{\rm
and}\,\,\,s_{1}(\kappa_{T}^{-1}(t))=s_{2}(t)\,\,\,{\rm for\,\,\,all}\,\,\,t\in
T_{2}.$ (e 5.14)

Recall that for the classification of a class of $C^{*}$-algebras, there must
be three ingredients:

(1) A definition of invariants for $C^{*}$-algebras in the class;
(2) A (range) theorem that says exactly what values of the invariants defined
in (1) occur for $C^{*}$-algebras in the class;
(3) the isomorphism theorem, i.e., the statement that any two $C^{*}$-algebras
in the class with the same, i.e., isomorphic invariants are isomorphic.

Let us now present a unified classification for finite separable amenable
simple ${\cal Z}$-stable $C^{*}$-algebras in the UCT class.

######  Definition 5.8.

From 2.15 and 2.19 of [43], with the pairing
$\rho_{A}:K_{0}(A)\to{{\operatorname{Aff}(\tilde{T}(A))}}$ defined in 5.4 (see
2.15 of [43]), the Elliott invariant for separable simple $C^{*}$-algebras
(see [18] and [20]), for the case ${{{\tilde{T}}(A)}}\not=\\{0\\},$ may be
described by the following sextuple:

${\rm{Ell}}(A):=((K_{0}(A),\Sigma_{\rho}(K_{0}(A)),{\tilde{T}}(A),\Sigma_{A},\rho_{A}),K_{1}(A)),$

where $\Sigma_{A}$ is a function in ${\rm LAff}_{+}(\tilde{T}(A))$ defined by

$\displaystyle\Sigma_{A}(\tau)=\sup\\{\tau(a):a\in{\rm
Ped}(A)_{+},\,\,\|a\|\leq 1\\}.$ (e 5.15)

Let $e_{A}\in A$ be a strictly positive element. Then
$\Sigma_{A}(\tau)=d_{\tau}(e_{A})=\lim_{\varepsilon\to
0}\tau(f_{\varepsilon}(e_{A}))$ for all $\tau\in{{\tilde{T}(A),}}$ which is
independent of the choice of $e_{A}.$ In this case, if
$\Sigma_{\rho}(K_{0}(A))$ has a maximum element $u$ and $\rho(u)=\Sigma_{A},$
then,

$\displaystyle\Sigma_{\rho}(K_{0}(A))=\\{g\in
K_{0}(A)^{+}:\rho(g)<\Sigma_{A}\\}\cup\\{u\\}$ (e 5.16)

(recall that $K_{0}(A)^{+}=\\{x\in
K_{0}(A):\rho_{A}(x)\in\operatorname{Aff}_{+}(\widetilde{T}(A))^{+}\\}\cup\\{0\\},$
see (e 5.11)). Otherwise,

$\displaystyle\Sigma_{\rho}(K_{0}(A))=\\{g\in
K_{0}(A)^{+}:\rho(g)<\Sigma_{A}\\}.$ (e 5.17)

In the case that $\widetilde{T}(A)=\\{0\\}$ (hence $\rho_{A}=0$), we also
define

$\displaystyle{\rm
Ell}(A)=\\{K_{0}(A),\Sigma_{\rho}(K_{0}(A)),{\widetilde{T}}(A),\Sigma_{A},\rho_{A}),K_{1}(A)),$
(e 5.18)

where $\Sigma_{A}=0.$ As in 5.7, since $\widetilde{T}(A)=\\{0\\},$ we have
$\\{x\in K_{0}(A)^{+}:\rho(x)<0\\}=\emptyset.$ Hence, in this case, when $A$
has a unit,

$\displaystyle\Sigma_{\rho}(K_{0}(A))=\\{x\in
K_{0}(A)^{+}:\rho(x)<0\\}\cup\\{[1_{A}]\\}=\\{[1_{A}]\\},$ (e 5.19)

and, when $A$ is not unital,

$\displaystyle\Sigma_{\rho}(K_{0}(A))=\\{x\in
K_{0}(A)^{+}:\rho(x)<0\\}=\emptyset.$ (e 5.20)

Recall that the set $\Sigma_{\rho}(K_{0}(A))$ is the distinguished set.

If $A$ is a separable ${\cal Z}$-stable $C^{*}$-algebra with
$\widetilde{T}(A)=\\{0\\},$ then by a result of M. Rørdam $A$ must be purely
infinite (Corollary 5.4 of [101]). In this case,

$\displaystyle{\rm Ell}(A)=(K_{0}(A),\\{[1_{A}]\\},\\{0\\},0,0,K_{1}(A)),$ (e
5.21) $\displaystyle{\rm or}\,\,\,$ $\displaystyle{\rm
Ell}(A)=(K_{0}(A),\emptyset,\\{0\\},0,0,K_{1}(A)).$ (e 5.22)

One may then simplify the invariant in these two cases by writing

$\displaystyle{\rm Ell}(A)=(K_{0}(A),[1_{A}],K_{1}(A)),\,\,\,{\rm
or}\,\,\,{\rm Ell}(A)=(K_{0}(A),\emptyset,K_{1}(A)).$ (e 5.23)

In other words, in the purely infinite simple case, $A$ has no unit if and
only if the distinguished set $\Sigma(K_{0}(A))=\emptyset.$ The definition
above gives the first ingredient of the classification.

Let $B$ be another separable $C^{*}$-algebra. A homomorphism $\Gamma:{\rm
Ell}(A)\to{\rm Ell}(B)$ consists of a homomorphism
$\Gamma_{0}:(K_{0}(A),\Sigma_{\rho}(K_{0}(A)),{\widetilde{T}}(A),\Sigma_{A},\rho_{A})$
to $(K_{0}(B),\Sigma_{\rho}(K_{0}(B)),{\widetilde{T}}(B),\Sigma_{B},\rho_{B})$
(as in 5.7, see (e 5.13) and (e 5.14) also) and a homomorphism
$\kappa_{1}:K_{1}(A)\to K_{1}(B)$. We say that $\Gamma$ is an isomorphism if
both $\Gamma_{0}$ and $\kappa_{1}$ are isomorphisms.

In the case that $\rho_{A}(K_{0}(A))\cap{\rm
Aff}_{+}(\widetilde{T}(A))=\\{0\\},$ we often consider the reduced case that
$T(A)$ is compact which gives a base for $\widetilde{T}(A).$ In that case, we
may write
${\rm Ell}(A)=(K_{0}(A),T(A),\rho_{A},K_{1}(A)).$ Note that, in the said
situation, $\Sigma_{\rho}(K_{0}(A))=\\{0\\}$ and $\widetilde{T}(A)$ is
determined by $T(A)$ and $\Sigma_{A}(\tau)=1$ for all $\tau\in T(A).$

Note, that in the special case that $K_{0}(A)=K_{1}(A)=\\{0\\},$ the Elliott
invariant may be simplified to $(\widetilde{T},\Sigma_{A}).$ Recall that a
$C^{*}$-algebra $A$ is $KK$-contractible if it $KK$-equivalent to $0.$ In
particular $K_{i}(A)=0$ ($i=0,1$). Before we discuss the general case, let us
first state the following theorem:

######  Theorem 5.9 (Elliott, Gong, Lin, and Niu, Theorem 7.5 of [27]).

The class of KK-contractible stably projectionless simple separable
C*-algebras with finite nuclear dimension is classified by the invariant
$(\widetilde{T}(A),\Sigma_{A}).$ Any $C^{*}$-algebra $A$ in this class is a
simple inductive limit of Razak algebras.

######  Definition 5.10.

Let $A$ be a finite separable is traditional to consider the following
distinguished subset of $K_{0}(A),$

$\displaystyle\Sigma(K_{0}(A))=\\{x\in K_{0}(A):x=[p]\,\,{\rm
for\,\,some\,\,projection},\,\,p\in A\\}.$ (e 5.24)

Moreover $K_{0}(A)_{+}=\\{x\in K_{0}(A):x=[p]\,\,{\rm
for\,\,some\,\,projection}\\}.$

Note that, for a finite exact separable simple ${\cal Z}$-stable
$C^{*}$-algebra $A,$ by a result of Rørdam (Corollary. 5.4 of [101]),
$\widetilde{T}(A)\not=\emptyset.$ For finite separable amenable simple
$C^{*}$-algebras, the next theorem should be regarded as an integral part of
the range theorem.

######  Theorem 5.11 (Gong-Lin, Theorem 5.2 of [43] and Corollary A.7 of
[26]).

Let $A$ be a finite separable amenable simple ${\cal Z}$-stable
$C^{*}$-algebra. Then

$\displaystyle K_{0}(A)_{+}=\\{x\in K_{0}(A):\rho(A)>0\\}\cup\\{0\\}\,\,\,{\rm
and}\,\,\,\Sigma(K_{0}(A))=\Sigma_{\rho}(K_{0}(A)).$ (e 5.25)

In other words,
$(K_{0}(A),\Sigma(K_{0}(A)),\widetilde{T}(A),\widetilde{\langle
e_{A}\rangle},\rho_{A})$ is a scaled simple ordered group pairing (see
Definition 5.7 above and Definitions 2.7 and 2.15 in [43]) (and $K_{1}(A)$ is
a countable abelian group).

The range theorem (the second ingredient) below was first described by Elliott
in [20] for finite case.

######  Theorem 5.12 (Gong-Lin, Theorem 5.3 of [43], Theorem 5.6 of [30], and
also 4.3 above).

Let $(G_{0},\Sigma_{\rho}(G_{0}),T,s,\rho)$ be a simple ordered group pairing
with distinguished set $\Sigma(G_{0})$ and $G_{1}$ be a countable abelian
group. Then there is a simple separable amenable ${\cal Z}$-stable
$C^{*}$-algebra $A$ which satisfies the UCT such that

$\displaystyle((K_{0}(A),\Sigma_{\rho}(K_{0}(A)),\widetilde{T}(A),\widetilde{\langle
e_{A}\rangle},\rho_{A}),K_{1}(A))=((G_{0},\Sigma_{\rho}(G_{0}),T,s,\rho),G_{1}).$
(e 5.26)

(i) In the case that $\widetilde{T}(A)\not=\\{0\\},$ A is unital if and only
if $G_{0}$ has a unit $u,$ i.e., $u$ is the non-zero maximum in
$\Sigma_{\rho}(K_{0}(A))=\Sigma(K_{0}(A))$ and $\rho_{A}(u)=s$ (see Definition
5.7). If $\rho(G_{0})\cap\operatorname{Aff}_{+}(T)\not=\\{0\\},$ then
$K_{0}(A)_{+}=\\{x\in K_{0}(A):\rho(x)>0\\}\cup\\{0\\},$ $A$ can be chosen to
have rational generalized tracial rank at most one (see Definition 4.2) and be
an inductive limit of subhomogeneous $C^{*}$-algebras of spectra with
dimension no more than $3.$ If
$\rho(G_{0})\cap\operatorname{Aff}_{+}(T)=\\{0\\},$ then A is stably
projectionless and A can be chosen to be locally approximated by
subhomogeneous $C^{*}$-algebras with spectra having dimension no more than
$3.$ If $G_{1}=\\{0\\}$ and $G_{0}$ is torsion free, then $A$ can be chosen to
be an inductive limit of 1-dimensional NCCW complexes with trivial $K_{1}.$

(ii) In the case that $\widetilde{T}(A)=\\{0\\},$ $A$ is purely infinite. $A$
is non-unital if and only if the distinguished set is empty, and unital if the
distinguished set $\Sigma_{\rho}(K_{0}(A))=\\{[1_{A}]\\}.$

Finally, let us present the isomorphism theorem:

###  5.2 The isomorphism theorem

We now present the third ingredient.

######  Theorem 5.13 (Gong and Lin, Theorem 14.9 of [44]).

Let $A$ and $B$ be finite separable amenable simple ${\cal Z}$-stable
$C^{*}$-algebras satisfying the UCT. Then $A\cong B$ if and only if

$\displaystyle((K_{0}(A),\Sigma(K_{0}(A)),{\widetilde{T}}(A),{{\widehat{\langle
e_{A}\rangle}}},\rho_{A}),K_{1}(A))$ (e 5.27) $\displaystyle\hskip
21.68121pt\cong((K_{0}(B),\Sigma(K_{0}(B)),{\widetilde{T}}(B),{{\widehat{\langle
e_{B}\rangle}}},\rho_{B}),{{K_{1}(B)}}).$ (e 5.28)

(1) If we restrict ourselves to non-unital $C^{*}$-algebras (i.e., we assume
that the $C^{*}$-algebras are non-unital), the invariant used in Theorem 5.13
can be simplified to

$((K_{0}(A),{\widetilde{T}}(A),{{\widehat{\langle
e_{A}\rangle}}},\rho_{A}),K_{1}(A)).$

This is because

$\displaystyle\Sigma(K_{0}(A))=\\{x\in K_{0}(A),x=[p]\,\,{\rm
for\,\,some\,\,projection}\,\,p\in A\\}$ (e 5.29) $\displaystyle=\\{x\in
K_{0}(A):x\in K_{0}(A)_{+}\,\,\,{\rm and}\,\,\,\rho_{A}(x)<{{\widehat{\langle
e_{A}\rangle}}}\\}$ (e 5.30)

which is determined by ${{\widehat{\langle e_{A}\rangle}}}.$

On the other hand, if we consider only the unital case, of course, the
invariant may be formulated as $(K_{0}(A),[1_{A}],T(A),\rho_{A},K_{1}(A)).$
However, we think the invariant set should work for both unital and non-unital
cases. One then should be able to tell from the invariant set itself whether a
simple $C^{*}$-algebra has a unit or not. The unified classification
statements of Theorem 5.13 and Theorem 5.12 achieve that goal.

Note that Theorem 5.12 states that all Elliott invariants satisfying the
description of Theorem 5.11 can be achieved by a separable amenable simple
${\cal Z}$-stable $C^{*}$-algebra in ${\cal N}$ (see Definition 2.14).
Omitting the UCT, but combining the purely infinite simple case (see [93] and
[51]), let us state the following theorem:

######  Theorem 5.14.

Let $A$ and $B$ be separable amenable simple ${\cal Z}$-stable
$C^{*}$-algebras in ${\cal N}.$ Then $A\cong B$ if and only if

$\displaystyle((K_{0}(A),\Sigma_{\rho}(K_{0}(A)),{\widetilde{T}}(A),{{\widehat{\langle
e_{A}\rangle}}},\rho_{A}),K_{1}(A))$ (e 5.31) $\displaystyle\hskip
21.68121pt\cong((K_{0}(B),\Sigma_{\rho}(K_{0}(B)),{\widetilde{T}}(B),{{\widehat{\langle
e_{B}\rangle}}},\rho_{B}),{{K_{1}(B)}}).$ (e 5.32)

Note that, in the purely infinite simple case, the Elliott invariants may be
simplified as

$\displaystyle{\rm Ell}(A)=(K_{0}(A),[1_{A}],K_{1}(A)),\,\,\,{\rm
or}\,\,\,{\rm Ell}(A)=(K_{0}(A),\emptyset,K_{1}(A)).$ (e 5.33)

On the other hand, if $A$ is known to be stable, then the Elliott invariant
may be simplified as

$\displaystyle{\rm Ell}(A)=(K_{0}(A),\widetilde{T}(A),\rho_{A},K_{1}(A))$ (e
5.34)

(finite or infinite).

The Elliott program of classification of separable amenable $C^{*}$-algebras
does not end at Theorem 5.11, Theorem 5.12 and Theorem 5.13. As mentioned
earlier, there are known simple AH-algebras which are not ${\cal Z}$-stable.
There is also a question whether non-simple ${\cal Z}$-stable $C^{*}$-algebras
could be classified, and if so, what would be the right formulation of the
invariant (see [47], [37], [38], [48], [10], [49], [35], [36], and [76]).
Therefore Theorem 5.13 is not even the beginning of the end. However, it does
classify the class of simple $C^{*}$-algebras which we have been mostly
interested in for decades. The simple separable amenable $C^{*}$-algebras
classified by Theorem 5.11, Theorem 5.12, and Theorem 5.13 are of lower rank,
or lower dimension (rank zero or one). A march to simple $C^{*}$-algebras of
higher rank has a long way to go ([111] and [107]). What is perhaps a modest
beginning was made recently in [28]. Nevertheless, Theorem 5.11, Theorem 5.12
and Theorem 5.13 summarize the decades of work and achievements by many people
in the Elliott program for simple amenable ${\cal Z}$-stable $C^{*}$-algebras.
It is, perhaps, not an exaggeration to state that these three theorems
(together with the classification of amenable purely infinite simple
$C^{*}$-algebras in [93]) represent the end of the beginning.

Finally, it is exciting to note that a new approach to the classification
program—and in particular to the finite case of Corollary 4.9—has recently
been developed ([6]).

Acknowledgements: This paper is partially supported by the Research Center for
Operator Algebras at East China Normal University in Shanghai which is
partially supported by Shanghai Key Laboratory of Pure Mathematics and
Mathematical Practice, Science and Technology Commission of Shanghai
Municipality (Grant No. 22DZ2229014). The first named author is supported by
National Natural Science Foundation of China (Grant No. 11920101001), and the
third named author is supported by a travel grant from the Simons Foundation
(Grant No. MP-TSM-00002606).

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Guihua Gong
University of Puerto Rico
San Juan, Puerto Rico
USA, Rio Piedras, Puerto Rico 00936
<EMAIL_ADDRESS>
Huaxin Lin
East China Normal University,
Shanghai, 200062 and
University of Oregon
Eugene, Oregon 97402
USA
<EMAIL_ADDRESS>
Zhuang Niu
University of Wyoming
Laramie, Wyoming, 82072
USA
<EMAIL_ADDRESS>