W.P. Grant and S. E. Ahnert

# Modular decomposition of protein structure using community detection

William P. Grant †
Theory of Condensed Matter Group  Cavendish Laboratory  University of
Cambridge  Cambridge  CB3 0HE.
†
Sebastian E. Ahnert
Theory of Condensed Matter Group  Corresponding author<EMAIL_ADDRESS>Cavendish Laboratory  University of Cambridge  Cambridge  CB3 0HE and
Sainsbury Laboratory  University of Cambridge  Cambridge CB2 1LR

###### Abstract

As the number of solved protein structures increases, the opportunities for
meta-analysis of this dataset increase too. Protein structures are known to be
formed of domains; structural and functional subunits that are often repeated
across sets of proteins. These domains generally form compact, globular
regions, and are therefore often easily identifiable by inspection, yet the
problem of automatically fragmenting the protein into these compact
substructures remains computationally challenging. Existing domain
classification methods focus on finding subregions of protein structure that
are conserved, rather than finding a decomposition which spans the full
protein structure. However, such a decomposition would find ready application
in coarse-graining molecular dynamics, analysing the protein’s topology, in de
novo protein design and in fitting electron microscopy maps. Here, we present
a tool for performing this modular decomposition using the Infomap community
detection algorithm. The protein structure is abstracted into a network in
which its amino acids are the nodes, and where the edges are generated using a
simple proximity test. Infomap can then be used to identify highly intra-
connected regions of the protein. We perform this decomposition systematically
across 4000 distinct protein structures, taken from the Protein Data Bank. The
decomposition obtained correlates well with existing PFAM sequence
classifications, but has the advantage of spanning the full protein, with the
potential for novel domains. The coarse-grained network formed by the
communities can also be used as a proxy for protein topology at the single-
chain level; we demonstrate that grouping these proteins by their coarse-
grained network results in a functionally significant classification.
community detection; protein structure; biological networks; spatial networks.

##  1 Introduction

All proteins are formed of chains of covalently bonded amino acids (also known
as residues). The pattern of non-covalent bonding between units of the chain
is what causes the protein to fold into its compact native structure;
specifying the sequence of amino acids in a protein is sufficient to uniquely
determine its folded shape [1]. This structure then allows the protein to
carry out its designated role within the cell.

Solving a protein’s structure is costly in time and effort, yet the number of
solved structures is growing rapidly. Over 130 000 protein structures are now
publicly available in the Protein Data Bank (PDB) [2], and the size of this
dataset is growing exponentially [3]. A widely-researched option for
extracting insight from this dataset involves the search for protein domains;
functional or structural subunits of a protein structure. Finding domains that
are conserved between proteins helps to elucidate the relationship between a
protein’s structure and its function in the cell, and to classify the proteins
into a taxonomy based upon their common structural features. The first efforts
to assign protein domains were based upon manual expert curation [4]. In
recent years, two alternative databases involving both manual curation and
computational assignment have emerged as mainstays; the CATH [5] and SCOPe [6]
databases. These databases focus on the domain as a structurally conserved
unit, rather than as a compact, globular substructure, and as such the SCOPe
and CATH labellings of the protein do not span the complete structure. Another
widely used tool is the PFAM database [7], which uses hidden Markov models to
discover conserved regions of protein sequence.

One plausible alternative definition of a domain is that of a community on a
protein structure network. Protein structure networks have been widely used in
which the protein’s amino acids are taken as the nodes of the network, with a
wide variety of approaches taken to generate the edges, often using proximity
of the $C_{\alpha}$ atoms (the central atom in each amino acid, bonded both to
the amino acid’s side chain and to the neighbouring amino acids via peptide
bonds) [8]. This abstraction has shown promise in analysing individual
proteins to identify key residues (amino acids) in allosteric communication
[9, 10, 11, 12] and protein thermal stability [13]. Tools have been developed
to assist with the creation and visualization of the networks [14, 15].

The community structure of protein structure networks has been previously
studied for individual proteins [16, 17, 18], showing that the community
structure often aligns well with intuitive functional domains. Other work [19,
20] has validated network-based clustering over more traditional spatial
clustering methods such as k-means clustering [21] and average-linkage
clustering [22].

However, previous network-based methods [19, 20] have yet to be scaled to the
set of proteins as a whole, possibly due to the computational cost involved.
In this work, we provide a comparison of network communities to known domain
assignments for a large set of distinct proteins (4000 non-redundant protein
chains). We offer an approach using the Infomap community detection method,
which uses the compression of a random walker’s movement on the network to
detect hierarchical community structure [23]. This notion of hierarchical
community structure is required in order to account for the known multi-scale
structure of proteins. We introduce a modified Jaccard measure to validate the
generated community structures, and investigate the coarse-grained networks
obtained by condensing each community into a single node, as a proxy for
protein architecture.

Non-network-based comprehensive studies of protein structure such as [22] only
compare the numbers of domains found, not the assignments of residue positions
to domains. Such approaches would therefore also not allow us to generate
condensed networks of modules, and ignore or discard information about the
hierarchical nature of community structure, for example by choosing a single
cut-off point for the clustering dendrogram [22].

##  2 Methods

The analysis consists of three steps: the generation of the network from the
protein structure, the community detection on the network and the storage and
analysis of the communities as regions of the protein.

###  2.1 Network generation

There are many plausible approaches to generating a network representation of
a protein’s structure. The nodes of the network could be either the protein’s
atoms [11] or residues [8]. For a residue network, the edges are generated if
two residues are within a certain distance. This distance measure can be based
upon the inter-$C_{\alpha}$ distance, the inter-$C_{\beta}$ distance, or on
the number of pairs of atoms within a certain proximity. Previous literature
[8, 14] has established a cut-off distance of $8\,\AA$ for $C_{\alpha}$ or
$C_{\beta}$ networks, and $\sim 5\,\AA$ for networks based on the number of
neighbouring atoms.

Here, a naïve yet flexible approach to network generation is used, which can
generate either atomic networks or residue networks as required. Given the
atomic positions from a PDB file, we let the atoms be nodes of the network.
Undirected edges are then generated between atoms that are closer than a given
cut-off distance. The cut-off distance between atoms $i$ and $j$ is defined as
$c_{ij}=s\left(r_{i}+r_{j}\right)$, where $r_{i}$ is the covalent radius of
atom $i$, and $s$ is a scaling parameter that can be varied to generate a
network with higher or lower connectivity as required. If an atomic network is
required, the edges are linearly weighted by proximity of the relevant atoms.
If a residue network is required, the network is condensed by letting the
amino acids be nodes in the network, with edges weighted according to the
number of neighbouring atoms in the original atomic network. In what follows,
residue networks with a value of $s=4$ are used, following [8].

Performing this analysis on a protein with multiple chains often results in a
network with distinct connected components, corresponding to each chain. As
such, for this analysis the proteins are first split by chain. This helps
ensure that any results are fixed at the sub-quaternary level.

Using a network generation tool written in Rust [24], PDB files containing 10
000 atoms can be parsed in this way in under 1 s.

###  2.2 Community detection

In choosing a community detection algorithm, we require a method that does not
require the length scale or number of communities to be specified beforehand;
we also require a method that is fast enough to allow for all 130 000 proteins
in the PDB to be analysed in a reasonable timeframe. We need the method to
detect hierarchical community structure, in order to investigate the multi-
scale structure of the protein, and a method with a resolution limit that will
not impede the discovery of domain-level structure. Infomap [23] satisfies
these constraints, along with known accuracy on benchmark graphs. Infomap has
the disadvantage that it is prone to overpartitioning networks with geometric
constraints, including spatial networks such as those generated in this work
[25]. However, empirically we see that the partitions generated correspond
well to the domain-level structure of the protein (see overleaf).

###  2.3 Storage

All networks, partitions and results are stored in a MongoDB database [26].
This prevents duplication of effort; for a given parameter set, the database
is first queried for the relevant information. If not found, then the relevant
calculation is performed and the results stored in the database. In this way a
large data set of protein structures with their community structure can be
acquired.

###  2.4 Performance

In order to compare the match between the structure found using community
detection and that found using other methods, we need a quantitative measure
of similarity [27]. Traditional performance metrics such as the Normalized
Mutual Information are unsuitable for this task; the predicted structure (for
instance the PFAM domain structure [7]) generally occupies only a subset of
the protein, whilst the generated community structure tiles the protein
completely. Extra structure outside the region spanned by the prediction
should not be penalized.

To this end, we modify the Jaccard index (JI), as follows. The JI is defined
as the intersection between two sets, divided by their union, where in this
case the sets correspond to regions of the protein sequence. This index is
modified as follows:

For each ‘expected’ domain:

  * •

Calculate the JI for all generated communities that overlap with the expected
domain, i.e. $\frac{A\cap B}{A\cup B}$, where $A\cap B$ is the size of the
overlap and $A\cup B$ the total length of sequence spanned by either the
expected domain or the generated community.

  * •

Perform an average of all the calculated JIs, weighted by the proportion of
the total expected domain spanned by each community.

This gives a score for each expected domain in the protein, indicating how
well it is reflected in the community structure. On test data, this modified
JI performs reasonably (see Fig. 1), giving high scores to close matches and
low scores to poor matches. Note that like the original JI [27], this score
does not take values in the full range $[0,1]$.

Figure 1: An illustration of the modified JI on example data. The expected
domains (for instance, PFAM domains) are given above, and example community
structures below for six possible cases. Each coloured block indicates a
domain, with grey indicating unannotated regions. The upper right example
shows a perfect score, as each PFAM domain matches a community perfectly. In
contrast, the upper left figure shows that one PFAM domain has been split into
two communities, giving the matching to that domain a score of 0.5 and an
average score for the total match of 0.75. Note that the lower right figure,
representing roughly the poorest imaginable case (a randomly shuffled two-
community partition), still achieves a modified JI of 0.2.

In order to calculate the significance of a given modified JI, we use the
z-score. This is defined as:

$z=\frac{\tilde{J}-\mu}{\sigma}$

Where $\tilde{J}$ is the modified JI between the expected and generated
partitions. $\mu$ and $\sigma$ are the average and standard deviation of the
modified JI between the expected partition and a set of null models. $\mu$
therefore indicates the modified JI expected by chance. A z-score of two
indicates that the modified JI between the generated and expected partitions
is two standard deviations higher than the expected value, and therefore,
corresponds to a p-value of $\sim 0.02$ (assuming a normal distribution).

These null models should be randomly generated, sharing some key properties of
the generated community structure. In this work, the null models are created
by constraining the number of boundaries (changes from one community to
another along the sequence), and the total number of communities. Boundaries
and community labels are then placed randomly to obey these constraints.
Figure 2 shows the community structure to be tested above, with six generated
null models below. These models succeed in capturing the rough features of the
generated structure, whilst preserving randomness.

(a)  (b)

(c)  (d)

Figure 2: Randomly generated null models, such that the number of boundaries
between communities, and the total number of communities, is preserved. The
generated structure to be tested is shown above, with six null models shown
below. These null models succeed in capturing the properties of the test
structure, ensuring that the comparison between null model and test is fair.

##  3 Results

Empirically, we see that a scaling parameter of approximately 4 gives
communities corresponding to compact, globular regions of the protein
structure (Fig. 3). We can quantify the extent to which these communities
overlap with known protein annotations using the z-score as defined
previously. Here, we test the correspondence between the known PFAM domains,
and the generated community structure. In general, there is significant
agreement, with the majority of proteins having a z-score greater than 2 (Fig.
4).

(a)

(b)

Figure 3: The generated community structure (a) using a residue network with
scaling parameter $s=4$ compared with the known SCOPe domains (b) for a
pyruvate kinase with PDB code 1PKN. One colour signifies one domain/community.
Here, we see that one of the communities matches well to the existing SCOPe
domain (both shown in red).

Figure 4: Histogram showing the z-score for the modified JI between the
generated community structure, and the PFAM domains, for $\sim$1000 test
proteins, showing that in many cases the agreement is extremely significant.

The communities found are based purely on the protein’s structure, whilst the
PFAM domains are based purely on sequence. As such, we expect discrepancies
when the PFAM sequence domains correspond to more spatially extended, less
well-connected regions of the structure. We can measure this by calculating
the conductance of the regions of the network responding to the PFAM domains.
If the set of nodes of a network $V$ is split into two subsets $S$ and
$\bar{S}$, the conductance is defined as:

$\phi(S)=\frac{\sum_{i\in S,j\in\bar{S}}A_{ij}}{\min(A(S),A(\bar{(}S))}$

Where $A_{ij}$ are the elements of the networks adjacency matrix, and
$A(S)=\sum_{i\in S}\sum_{j\in V}A_{ij}$. Hence $\phi(S)\in[0,1]$, with a lower
conductance corresponding to a more isolated region of the network. We expect
the modified JI and the conductance to negatively correlate; Figure 6 shows
this is indeed the case.

We can compare the communities generated using Infomap to previous network-
based attempts to assign domains, which used correlation networks and a
modularity-based method [20]. Figure 6 compares these results qualitatively to
SCOPe annotations and to the results obtained using our protocol. Figure 7
compares the results quantitatively, using the z-score. A drawback of the
correlation-based approach is that a set of homologous proteins is needed; our
method has the advantage that it can be performed on single proteins, meaning
that the partition spans the full protein structure, and making the approach
scalable to larger datasets.

Figure 5: The conductance of the PFAM domain, when mapped onto the network,
against the modified JI (indicating how well it corresponds to the community
structure) The conductance is 0 for perfectly-isolated communities, and 1 for
communities fully connected to the rest of the network, so we expect a
negative correlation between modified JI and conductance; this is seen for the
proteins studied here.

(a)

(b)

(c)

(d)

Figure 6: A comparison of annotations of the protein 1BF2. (a) The
decomposition generated in previous work using a modularity-based method,
combined with residue correlation analysis [20]. Dark blue regions correspond
to unannotated regions of the structure. (b) The decomposition using Infomap
presented in this article. (c) The domains listed in the SCOPe structural
domain database. (d) The same comparison, along with the PFAM annotations,
presented as labellings of the protein sequence. Again, dark blue represents
unannotated regions of the sequence.

Figure 7: A comparison of the z-scores for a set of 20 reference proteins,
giving the significance of the overlap between SCOP (above) and PFAM (below)
for the modularity + correlation method [20] and the proposed Infomap-based
method. We see that in both cases the Infomap-based method has a more
significant similarity to existing annotations.

In addition to the communities’ potential value as structural domains, the
arrangement of the communities may be used as a proxy for topology. The
community structure can be converted to a coarse-grained network in which the
protein’s communities become nodes, linked if the respective communities are
neighbours. We can then classify the proteins according to the arrangement of
their communities, by grouping proteins with isomorphic coarse-grained
networks.

If the community structure is truly capturing the protein’s topology, we
expect this grouping to reveal aspects of protein function. We can test this
claim using Gene Ontology (GO) term analysis [28]. This effort assigns
functional relevance (e.g. lactase activity, oxidoreduction) to genes. The
SIFTS project [29] maps these GO terms to records in the PDB, meaning that
each protein now has a set of labels encoding information about its function
in the cell. We can then test if the grouping results in enriched GO terms,
i.e. terms appearing more often than expected by chance [30, 31]. For $N$
total proteins, and a subset of that dataset with $n$ proteins, the
probability of a GO term being found is given by the cumulative distribution
function (CDF) of the hypergeometric function. For a given GO term, let $k$ be
the number of times it occurs in the subset, and $K$ be the number of times it
occurs across the full dataset. Then the likelihood that the term would be
seen $k$ times by chance is:

$CDF=1-\frac{{n\choose k+1}{N-n\choose K-(k+1)}}{{N\choose
K}}\;{}_{3}F_{2}\left[\begin{matrix}1,\>\>\>&&k+1-K,&&k+1-n\\\
&k+2,&&N+k+2-K-n&\end{matrix}\;\;;1\right]$

Where ${}_{3}F_{2}$ is the generalized hypergeometric function. From the CDF,
we can acquire p-values for a given grouping and GO term; we consider GO-terms
with a p-value of less than 0.01 to be enriched in the subset to a
statistically significant extent. As we testing $M$ distinct GO terms, we
account for multiple hypothesis testing by applying the Bonferroni correction.
If comparisons of $M$ GO terms are being made, the raw p-value is multiplied
by $M$ to give a more conservative estimate of the likelihood.

We see that 90% of the proteins studied can be represented by only 10 coarse-
grained networks, all of which are associated with GO-term enrichment (Fig. 8
and Table 8).

Figure 8: The coarse-grained networks generated from the community structures
of approximately 4300 protein chains, taken randomly from a non-redundant
subset of the Protein Data Bank. Only coarse-grained networks common to at
least three proteins are shown (accounting for $\sim 3900$ proteins in total).
The node size is proportional to the number of proteins exhibiting that
coarse-grained network.

Coarse-grained network | Number of enriched GO terms (p<0.01) | Number of proteins
---|---|---
| 331 | 1725
| 130 | 307
| 116 | 841
| 104 | 445
| 34 | 55
| 52 | 207
| 48 | 26
| 23 | 19
| 9 | 6
| 22 | 8

tableThe ten most common protein topologies in the data set studied, ordered
by prevalance.

##  4 Conclusion

There have been many attempts to define the domain, as a compact, repeated
unit of protein structure. But choosing these compact, globular substructures
in an automated way has traditionally been challenging. We present results
showing that a simple weighted network of residue contacts analysed with
Infomap can successfully fragment a protein into compact modules. By using a
modified JI, we show that in general these modules correlate well with
existing PFAM annotations, yet have the advantage that they span the full
protein structure. This has potential applications in molecular dynamics and
electronmicroscopy.

We also show that by generating a coarse-grained network, in which the
communities of the network are taken as nodes, we can group a large set of
proteins in a way that gives significant functional enrichment, as measured by
the prevalence of GO terms. This suggests that the community structure can be
used as a proxy for the protein topology.

The next step will be to use this approach to search for repeated communities
with similar internal topology that have not yet been identified as domains,
with the hope of establishing a new framework for domain discovery.

## Funding

This work was supported by the Engineering and Physical Sciences Research
Council Centre for Doctoral Training in Computational Methods Materials
Science [EP/L015552/1] to W.P.G. and the Royal Society and the Gatsby
Foundation to S.E.A.

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section2em3em subsection5em3.5em subsubsection8.5em4em section2em3em
subsection5em3.5em subsubsection8.5em4em

# A Reconstruction of
Algebraic Quantum Theory

Bharath Ron

###### Abstract

We obtain a condensed reconstruction of algebraic quantum theory, emphasizing
its foundational aspects and algebraic structure. We obtain the
$W^{*}$-algebra structure from elementary assumptions about observers and how
they can observe reality. This paper highlights the need for $W^{*}$-algebra
structure by directly obtaining the mathematical axioms from simple thought
experiments.

Keywords: algebraic quantum theory, reconstruction, $W^{*}$-Algebras

Every scientific theory comprises of a _linguistic representation_ of a
collection of postulates speculating the relations between facts of the world.
The linguistic structures enable predictions about facts of the world by
leveraging the postulated relations between the facts. The predictive
capabilities of a scientific theory can be made more reliable by refining its
faulty postulates. In this case we can only modify or remove the faulty
postulates. Such refinements lead to theories with fewer postulates which are
easier to validate and provide more reliable predictions. Some of these faulty
postulates might be embedded within the linguistic structures of the language
used for postulate representation. To circumvent such linguistic issues, we
may retain only those structures common to all languages. Such universally
common linguistic structures are _the rules of logic_ and hence we can view
_mathematics as the universal language of postulate representation_. By using
mathematics for postulate representation physics effectively avoids issues
arising from suboptimal linguistic structures. The reliability of predictions
of physical theories depends solely on its postulates. Formulation of physical
theories usually involves finding an appropriate mathematical theory where its
postulates _can_ be represented. If some of the structures of the mathematical
theory are unnecessary for postulate representation, then such structures can
constrain predictions that end up conflicting with observed facts. For
example, commutativity in classical theories leads to conflicts with observed
facts. Hence, it is crucial to figure out which assumptions of the
mathematical theory are _necessary_ for representation of postulates. This
necessity condition establishes an equivalence between the set of assumptions
of the mathematical theory and the set of postulates of the corresponding
physical theory. We say such a mathematical theory the _universal theory_ for
the postulates.

Our objective is to develop the universal mathematical theory for the
postulates we only consider the structures essential for postulate
representation. The study of such peculiar structures may contribute to the
development of new mathematics, which might be useful in the future. It is
however not our focus here. We show the necessity of $W^{*}$-algebra structure
for the postulates about how we observe the world around us, and hence may be
viewed as a reconstruction of quantum theory.

## Generalised Probabilistic Theories

The physical systems of interest in quantum theories are those whose observed
phenomena can be described by the notion of an _observable_. The physical idea
here is that the world can be described by sentences of the form _the
observable $\Sigma$ has the value $O$_. The next step is to articulate this
idea in mathematical language. This means that we need to construct an
appropriately predictive mathematical theory where sentences like the one
above can be represented. Hence the starting point is the construction of a
universal representation for this postulate. The construction of such a
universal representation for the postulate we use here is closest to the
instrumentalist approach, see [4],[5],[6], with some ideas borrowed from the
quantum logic approach, see [2],[3],[8],[13],[14], and algebraic quantum
theory. This can be thought of as a modern formulation of Heisenberg’s
original idea. The starting point is the notion of measuring instruments and
preparation instruments and since the only way to verify facts about the world
is by performing measurements in the world, the approach is very general. Any
changes occurring in the instruments during _measurements_ will be accepted as
objective outcomes. From this point of view, the fundamental notions of
quantum theory will have to be defined operationally in terms of measuring
instruments, preparation procedures, and the prescriptions for their
application. The theory is then interpreted entirely in terms of such
instruments and outcomes which are the changes occurring to instruments.

Every preparation procedure is characterised by the kind of system it
prepares. The measuring instruments are capable of undergoing upon performing
a measurement evaluating the truth value of collection of possible outcomes.
The possible results of such an experiment are called outcomes of the
experiment and the observable change in the instrument is called an effect. To
simplify the procedure consider instruments that record _hits_. These
instruments perform simple _yes-no_ measurements. Any measurement can be
interpreted as a combination of _yes-no_ measurements. These _yes-no_
instruments can be used to build any general instrument. Suppose we have such
a measuring instrument, label its measuring instrument by $O$. If the
experiment is conducted a lot of times, we get a relative frequency of
occurrence of _yes_. Here _yes_ is an observable change in the instrument. It
is hence an observable effect. To every preparation procedure $\rho$ and
measuring instrument $O$ there exists a probability $\mu(\rho|O)$ of the
occurrence of _yes_ associated with the pair,

$(\rho,O)\longrightarrow\mu(\rho|O).$

The numbers $\mu(\rho|O)$ are called operational statistics.

Two completely different preparation procedures may give the same operational
statistics, that is, they may give the same probabilities for every
experiments. Such preparation procedures must be considered equivalent
operationally. An equivalence class of preparation procedures yielding the
same operational statistics for experiments is called an ensemble. Similarly,
there may be two measuring instruments that have the same chances of
undergoing a change for similarly prepared systems. Such measuring instruments
must be operationally considered to be equivalent. An equivalence class of
change for measuring instruments is called an effect. An effect is the
equivalence class of all instruments that undergo a change for the same
possible outcome. By considering equivalence classes we have obtained the
structure of sets. Since we are considering equivalence classes, the
construction and behaviour of the instruments are irrelevant to the
discussion. We denote the set of ensembles by $\mathfrak{S}$ and the set of
effects by $\mathfrak{E}$. The pairing describing the operational statistics
is

$\mathfrak{S}\times\mathfrak{E}\xlongrightarrow{\mu}[0,1].$

For a possible outcome $O$, we will denote the corresponding effect by
$\mu(\cdot|{O})$. Each effect acts on the ensembles of the system, and each
ensemble acts on the effects of the system to yield the corresponding
operational statistics

$\displaystyle\mu\big{(}\cdot|\>{O}\big{)}$
$\displaystyle:\>\rho\mapsto\mu(\rho|O),$
$\displaystyle\mu\big{(}\>\rho\>|\cdot\big{)}$
$\displaystyle:\>O\mapsto\mu(\rho|O).$

The sets $\mathfrak{E}$ and $\mathfrak{S}$ together with the pairing
$\mu(\cdot|\cdot)$ will be called an operational theory. Since any two
preparations giving the same result on every effect represent the same
ensemble and two measurement procedures that cannot distinguish ensembles
represent the same effect. Therefore ensembles and effects must mutually
separate each other respect to $\mu(\cdot|\cdot)$.

If we view the ensemble as a representation of the experimenter’s beliefs
about the universe, the collection of effects allow the experimenter to verify
his beliefs about the universe. The results of experiments represent an
updating of the experimenter’s beliefs. If we define an observer as an
equivalence class of experimenters with the same set of beliefs, then we may
view operational theories _as the theory of observers and the observed._ The
set of ensembles corresponds to the set of all beliefs the observer _can_
hold, and represents the observer’s _subjective reality_. Since the space of
effects is the only link between an experimenter and objective reality, we may
identify the space of effects with objective reality itself. From this point
of view, we may view operational theories as the theory of duality between the
subjective and the objective.

## Locally Convex Structure

In order to be able to work with effects and ensembles we now embed them
inside sets with mathematical structure that respects the expected operational
relations. We now describe how the collection of effects and the collection of
ensembles inherit mathematical structures. In our case the operational
requirement is that we should be able to make sense of taking mixtures of
effects and ensembles. Suppose we can prepare systems in different ensembles
by varying settings of a preparation procedures, then by adjusting the
settings we should be able to prepare systems in mixtures of ensembles. We
expect the space of ensembles to have the structures to describe mixtures.

Let us assume that we can form such mixtures by changing the settings on the
preparation procedures. If such adjustments are parametrised by real numbers,
such as rotating a knob, then the notion of convexity needed for formalising
comes from the space of real numbers. The space of real numbers is a
convenient space because its convexity structure is related to a lot of other
structures, such as its topology, its vector space structure, and so on. As
discussed in [1], such a notion of mixing corresponds mathematically to the
notion of convex combination. Hence real vector space structure acts as a good
starting point. Hence we may expect the set $\mathfrak{S}$ to have the
structures for making sense of convexity and we expect that each functional
$\mu(\cdot|O)$ preserves the convex structure since we expect the operational
statistics to preserve this convexity. To make sense of taking convex linear
combinations we need to be able to make sense of linear combinations. Hence we
must embed the set of effects and the set of ensembles in a vector space with
respect to a field that is at least as large as the real numbers. It is
important to note that preparation procedures and measuring instruments
producing the same ensembles and effects are not equal, in fact, the notion of
equality will not even make sense. The transition from preparation procedures
and measuring instruments to ensembles and effects is a transition from the
real physical world to the abstract mathematical world. It should also be
noted that it does not make sense to _prepare_ closed systems, one has to
assume such systems start off in some state a priori.

A generalised probabilistic theory is an embedding of $\mathfrak{S}$ and
$\mathfrak{E}$ inside vector spaces, such that the ensembles and effects are
uniquely determined by the operational statistics they produce. This
uniqueness of operational statistics is called the principle of tomography in
quantum foundations and quantum information literature. To construct a
generalised probabilistic theory for $\mathfrak{S}$ and $\mathfrak{E}$ we
start by viewing effects and ensembles as linear functionals on a suitable
space. Denote by $\mathcal{A}_{*}$ the set of maps

$\mu(O)=\sum_{i\in F}\lambda_{i}\mu\big{(}\rho_{i}|O\big{)},$

for every $O$ in $\mathfrak{E}$, where $\\{\rho_{i}\\}_{i\in F}$ is a finite
collection of ensembles and $\lambda_{i}$ are scalars. Since the notion of
convexity is inherited from the notion of convexity on the real line, we
assume that the field $\mathbb{K}$ contains the field of real numbers. The
function $|\cdot|:\mathbb{K}\to\mathbb{R}^{+}$ denotes the standard modulus
function on $\mathbb{K}$. Any linear combination of such maps also belongs to
$\mathcal{A}_{*}$, it follows that $\mathcal{A}_{*}$ is a $\mathbb{K}$-vector
space. Denote by $\mathcal{E}$ the set of maps,

$E(\rho)=\sum_{i\in F}\lambda_{i}\mu\big{(}\rho|O_{i}\big{)},$

for every $\rho$ in $\mathfrak{S}$, where $\\{O_{i}\\}_{i\in F}$ is a finite
collection of effects and $\lambda_{i}$ are scalars. It follows that
$\mathcal{E}$ is also a $\mathbb{K}$-vector space. We embed ensembles inside
$\mathcal{A}_{*}$ with the map

$\displaystyle\rho\mapsto\mu\big{(}\rho\>|\cdot\big{)}.$

Similarly, we embed effects in $\mathcal{E}$ with the map

$\displaystyle O\mapsto\mu\big{(}\cdot|\>O\big{)}.$

The elements of the convex subsets $\mathfrak{E}$ of $\mathcal{E}$ and
$\mathfrak{S}$ of $\mathcal{A}_{*}$ are called effects and states
respectively.

The vector space structure of $\mathcal{A}_{*}$ and $\mathcal{E}$ allows us to
develop algebraic tools for studying effects and ensembles. So we embedded an
operational theory by viewing the effects and ensembles as linear functionals
on each other and took _quotients_ with operational equivalences. Since we
expect the ensembles and effects to be determined by the probabilities they
produce we must also expect the vector spaces $\mathcal{E}$ and
$\mathcal{A}_{*}$ to inherit a relation between each other from the pairing
$\mu(\cdot|\cdot)$ of $\mathfrak{E}$ and $\mathfrak{S}$. So we must have a
pairing

$\big{\langle}\cdot|\cdot\big{\rangle}_{\mathcal{E}}:\mathcal{A}_{*}\times\mathcal{E}\to\mathbb{K},$

Such that, for every preparation procedure $\rho$ and measurement outcome $O$,

$\Big{\langle}\mu(\rho|\cdot)\>\big{|}\>\mu(\cdot|O)\Big{\rangle}_{\mathcal{E}}=\mu(O|\rho)$

Hence we obtain a pairing of locally convex spaces equipped with the weak
topologies. We now have access to the theory of conjugate duality of locally
convex spaces.

## Banach Space Structure

The operational requirements force the space of effects and ensembles to be
vector spaces. The vector space structure is however insufficient for doing
analysis with effects and ensembles which would allow the theory to be
predictive. We now discuss how to introduce additional mathematical structures
on the space of effects and ensembles, based on operational heuristics. We
will later discuss the mathematically precise formulations of the heuristics.

Suppose we have two measuring instruments for an outcome where one is more
accurate than the other. In such a case, the less accurate measuring
instrument will readily undergo a change compared to the more accurate
instrument. If the instrument is too accurate, it becomes difficult to observe
any changes occurring in the measuring instrument. In this sense, the more
accurate instrument should be _closer_ to the instrument which never undergoes
any changes. So, we must expect $\mathcal{E}$ to have some structure that
quantifies how readily an instrument undergoes a change. The instrument which
always undergoes a change should be the easiest to notice changes and the
instrument which never undergoes any changes should be the hardest. Suppose we
have two measuring instruments which undergo changes for outcomes $O$ and
$O^{\prime}$, and let $\mu(\cdot|O)$ and $\mu(\cdot|O^{\prime})$ be the
corresponding effects. For combined instruments, we expect the accuracy change
for the combined instruments to depend on the accuracy of the component
instruments. Hence the changes in accuracy must be expected to vary
proportionally to the changes in accuracy of its components. This gives rise
to continuity requirements on the space of effects. We assume the existence of
the unit $1$, which corresponds to the instrument which is always true for any
preparation procedure. The unit must be unique since any other such element
must be operationally equivalent. Operationally, the unit corresponds to the
_existence_ of the system itself, since it must always be true. Similarly we
assume the existence of the unique $0$ element which outputs _no_ for every
preparation procedure, and can be thought of as a faulty instrument which
never undergoes any changes. Note that the less accurate instrument undergoes
a change for more events than intended. A faulty instrument never undergoes
any change, for any event. It has no possibility to be inaccurate, and hence
correspond to the most accurate instruments.

For any collection of instruments for a given outcome we can obtain a relation
between such instruments by their accuracy. Consider two instruments which can
undergo changes under some _similar_ situations. This means that we may have
some redundancies, or common inaccuracies, that is, they can measure some
common possible outcomes, or alternatively, they have differing accuracy for
measuring the common outcome. If the accuracy of either of the instruments is
changed, the accuracy of the combined instrument should also change, and this
change should be proportional to the change in accuracy of the instrument.

The notion of accuracy of instruments must give rise to a function on the
space of effects. In our case, it is easier to model the inaccuracy. If
changes in accuracy are modelled using the addition operation on the real
line, then the notion of inaccuracy must correspond to a real-valued function
$f$ such that,

$f\big{(}\mu(\>\cdot\>|\>O\>)+\mu(\>\cdot\>|\>O^{\prime}\>)\big{)}\leqq
f\big{(}\mu(\>\cdot\>|\>O\>)\big{)}+f\big{(}\mu(\>\cdot\>|\>O^{\prime}\>)\big{)}.$

Such functions are called sub-linear maps. The notion of inaccuracy of
instruments naturally gives rise to triangle inequality like properties.

We can describe this _inaccuracy function_ more conveniently as a supremum.
Since inaccuracy represents the maximal likelihood of undergoing a change, it
must correspond to the supremum of undergoing a change over all preparation
procedures. That is to say, the accuracy in the real world can only depend on
_reality_ which we expect to be closely related to preparation procedures.
This gives us a definition in terms of supremum over preparation procedures,
and can be defined on all of $\mathcal{E}$,

$\big{\|}E\big{\|}_{\mathcal{E}}\coloneqq\sup_{\omega\in\mathfrak{S}}\Big{[}\big{|}\big{\langle}E|\omega\big{\rangle}_{\mathcal{E}}\big{|}\Big{]},$

for every $E$ in $\mathcal{E}$, and the supremum is taken over all preparation
procedures and which corresponds to the convex embedded subset $\mathfrak{S}$
of $\mathcal{A}_{*}$. It immediately follows that $\|\cdot\|_{\mathcal{E}}$
satisfies the triangle inequality, and scaling condition,

$\displaystyle\|E+F\|_{\mathcal{E}}$
$\displaystyle\leqq\|E\|_{\mathcal{E}}+\|F\|_{\mathcal{E}}.$
$\displaystyle\big{\|}\lambda E\big{\|}_{\mathcal{E}}$
$\displaystyle=|\lambda|\big{\|}E\big{\|}_{\mathcal{E}},$

for all $E,F\in\mathcal{E}$ and $\lambda$ is a scalar. By definition of the
norm we have $\|1\|_{\mathcal{E}}=1$. By the assumption that there will always
exist some preparation procedure with non-trivial operational statistics, we
deduce that $\|\cdot\|_{\mathcal{E}}$ is non-trivial, that is,
$\|E\|_{\mathcal{E}}=0$ if and only if $E\equiv 0$. Hence
$\|\cdot\|_{\mathcal{E}}:\mathcal{E}\to\mathbb{R}$ defines a norm on
$\mathcal{E}$.

Similarly, $\mathcal{A}_{*}$ also inherits operator norm from $\mathcal{E}$.
We can have two preparation procedures, where the preparations settings are
slightly varied. We can say two ensembles are close to each other if the
operational statistics are close to each other for all effects. Hence how
similar two preparation procedures are can also be described as a supremum,
and this can be described on all of $\mathcal{A}_{*}$.

$\big{\|}\omega\big{\|}_{\mathcal{A}_{*}}\coloneqq\sup_{E\in\mathfrak{E}}\Big{[}\big{|}\langle
E|\omega\rangle_{\mathcal{E}}\big{|}\Big{]},$

for every $\omega$ in $\mathcal{A}_{*}$ and the supremum is taken over all
possible measurements which corresponds to the convex embedded subset
$\mathfrak{E}$ of $\mathcal{E}$. We can hence require both $\mathcal{E}$ and
$\mathcal{A}_{*}$ to be normed vector spaces.

The notion of limits is a mathematical notion, and by assumption must be
common to every experimenter, since we expect the rules of logic to be a
universal property of languages. This ensures that the predictions obtained by
an experimenter assuming the existence of limits must remain valid for any
other experimenter. This allows the experimenter to mathematically analyse and
predict facts about the world based on the experimentally verified beliefs.
Hence we may assume that the normed spaces $\mathcal{E}$ and $\mathcal{A}_{*}$
are complete with respect to their respective norms, and $\mathfrak{E}$ and
$\mathfrak{S}$ are closed convex subsets of $\mathcal{E}$ and
$\mathcal{A}_{*}$ respectively. The spaces $\mathcal{E}$ and $\mathcal{A}_{*}$
are generated by limits of linear combinations of elements of $\mathfrak{E}$
and $\mathfrak{S}$ respectively. By the boundedness of operational statistics
we note that effect and ensembles give rise to norm bounded linear functionals
on $\mathcal{A}_{*}$ and $\mathcal{E}$ respectively. By assumption
$\mu(\cdot|\cdot)$ separates $\mathfrak{S}$ and $\mathfrak{E}$. Since
$\langle\cdot|\cdot\rangle_{\mathcal{E}}$ coincides with $\mu(\cdot|\cdot)$
for ensembles and effects, it follows by the density of the span of
$\mathfrak{E}$ that

$\big{\langle}\cdot|\cdot\big{\rangle}_{\mathcal{E}}:\mathcal{A}_{*}\times\mathcal{E}\to\mathbb{K},$

is a non-degenerate pairing, and hence
$\langle\mathcal{E}|\mathcal{A}_{*}\rangle_{\mathcal{E}}$ must be a dual pair.
If $(\mathcal{A}_{*})^{*}$ is the space of norm bounded linear functionals on
$\mathcal{A}_{*}$ and as Banach spaces it follows that

$\mathcal{E}\cong\big{(}\mathcal{A}_{*}\big{)}^{*}$

This gives us access to the tools of duality theory of Banach spaces for
making predictions.

## Banach Algebra Structure

We expect varying accuracy of either of the measuring instruments
corresponding to $E$ or $F$ to vary the accuracy of $EF$ accordingly. Hence we
expect the algebraic structure on $\mathcal{E}$ to respect the topological
structure on $\mathcal{E}$ coming from the norm $\|\cdot\|_{\mathcal{E}}$. We
may perform a measurement before or after performing some other measurement.
This composite measurements can itself be thought of as a measuring
instrument. Heuristically, when these concatenated measurements have
inaccuracies which are independent of each other, we must expect the accuracy
of the concatenation, to amplify the inaccuracies so the total inaccuracy
would be that of the first instrument _and_ the second instrument and hence
must correspond to the product of inaccuracies. This can only be the case when
the accuracies of the two instruments are independent. Hence the accuracy of
the concatenation must be expected to be less than the product of individual
accuracies. This gives rise to an algebra structure on the space of effects
(note that there is no postulate so far _requiring_ commutativity of the
product structure). The space $\mathcal{E}$ comes equipped with a natural left
and right actions of effects on it given by

$\mathcal{L}_{E}F=EF\>\>;\>\>\mathcal{R}_{F}E=EF.$

We now study the topological properties of the product.

The composite element $\mathcal{L}_{E}F=EF$ corresponds to a well-defined
measuring instrument, and the accuracy of the combined instrument is given by
$\|EF\|_{\mathcal{E}}=\sup_{\omega\in\mathfrak{S}}|\langle
EF|\omega\rangle_{\mathcal{E}}|$. Since we are allowed to apply $E$ after
applying any other measuring instrument the accuracy of left-composition by
$E$ is given by,

$\displaystyle\big{\|}\mathcal{L}_{E}\big{\|}_{\mathcal{B}(\mathcal{E})}=\sup_{F\in\mathfrak{E}\atop\omega\in\mathfrak{S}}\Big{[}\big{|}\langle
EF|\omega\rangle_{\mathcal{E}}\big{|}\Big{]}=\sup_{F\in\mathfrak{E}}\bigg{[}\sup_{\omega\in\mathfrak{S}}\Big{[}\big{|}\langle
EF|\omega\rangle_{\mathcal{E}}\big{|}\Big{]}\bigg{]}=\sup_{F\in\mathfrak{E}}\Big{[}\big{\|}EF\big{\|}_{\mathcal{A}}\Big{]}.$

Similarly, the norm of $\mathcal{R}_{F}$ is given by,

$\big{\|}\mathcal{R}_{F}\big{\|}_{\mathcal{B}(\mathcal{E})}=\sup_{E\in\mathfrak{E}}\Big{[}\big{\|}EF\big{\|}_{\mathcal{E}}\Big{]}.$

So, both $\mathcal{L}_{E}$ and $\mathcal{R}_{F}$ belong to the space
$\mathcal{B}(\mathcal{E})$ of bounded operators on $\mathcal{E}$.

Since we expect the accuracy of the composite to change proportionally to the
change in accuracy of the composing instrument, we expect both
$\mathcal{L}_{E}$ and $\mathcal{R}_{F}$ to be continuous. This must relate the
norm $\|\cdot\|_{\mathcal{B}(\mathcal{E})}$ and the norm
$\|\cdot\|_{\mathcal{E}}$. To relate the two norms we start with the following
relation; $\mathcal{R}_{F}E=EF=\mathcal{L}_{E}F.$ By continuity of
$\mathcal{R}_{F}$ we have,
$\|\mathcal{R}_{F}E\|_{\mathcal{E}}=\|\mathcal{L}_{E}F\|_{\mathcal{E}}\leqq\|\mathcal{L}_{E}\|_{\mathcal{B}(\mathcal{E})}\>\|F\|_{\mathcal{E}}.$
Hence $\\{\mathcal{R}_{F}E\\}_{F\in\mathfrak{E}}$ is a bounded set for every
$E$.

Assuming $\mathcal{E}$ is a Banach space, by the Banach-Steinhaus theorem or
the uniform boundedness principle, point-wise boundedness of the set
$\\{\mathcal{R}_{F}E\\}_{F\in\mathfrak{E}}$ implies that
$\\{\mathcal{R}_{F}\\}_{F\in{\mathfrak{E}}}$ is uniformly bounded, with the
uniform bound
${\left|\kern-0.5382pt\left|\kern-0.5382pt\left|\mathcal{R}\right|\kern-0.5382pt\right|\kern-0.5382pt\right|}$.
Hence for every $E\in\mathcal{E}$,

$\big{\|}\mathcal{R}_{F}\big{\|}_{\mathcal{B}(\mathcal{E})}\leqq{\left|\kern-0.5382pt\left|\kern-0.5382pt\left|\mathcal{R}\right|\kern-0.5382pt\right|\kern-0.5382pt\right|}\>\big{\|}F\big{\|}_{\mathcal{E}},$

The map $\mathcal{R}$ is a bounded linear map,

$\displaystyle\mathcal{R}:\>$
$\displaystyle\mathcal{E}\to\mathcal{B}(\mathcal{E})$ $\displaystyle
F\to\mathcal{R}_{F}.$

The mapping $\mathcal{R}$ continuously maps the Banach space ${\mathcal{E}}$
into the Banach algebra $\mathcal{B}({\mathcal{E}})$ of bounded linear maps on
$\mathcal{E}$. The mapping $\mathcal{R}$ is a continuous algebra homomorphism.
Assuming $\|1\|_{\mathcal{E}}=1$, we have,

$\big{\|}E\big{\|}_{\mathcal{E}}=\big{\|}\mathcal{R}_{E}(1)\big{\|}_{\mathcal{E}}\leqq\big{\|}\mathcal{R}_{E}\big{\|}_{\mathcal{B}(\mathcal{E})}\leqq{\left|\kern-0.5382pt\left|\kern-0.5382pt\left|\mathcal{R}\right|\kern-0.5382pt\right|\kern-0.5382pt\right|}\>\big{\|}E\big{\|}_{\mathcal{E}}.$

We also have $\|\mathcal{R}_{1}(1)\|=\|1\|=1$, hence it follows that
${\left|\kern-0.5382pt\left|\kern-0.5382pt\left|\mathcal{R}\right|\kern-0.5382pt\right|\kern-0.5382pt\right|}=1$.

Hence $\mathcal{R}(\mathcal{E})$ is a norm-closed sub-algebra of
$\mathcal{B}({\mathcal{E}})$. The map $\mathcal{R}$ is an algebraic
isomorphism from ${\mathcal{E}}$ to $\mathcal{R}({\mathcal{E}})$. Since for
any two bounded linear operators $S$ and $T$ on ${\mathcal{E}}$ we have,
$\|TS\|\leqq\|T\|\>\|S\|$, $\mathcal{R}({\mathcal{E}})$ inherits this property
and we have,

$\big{\|}EF\big{\|}_{\mathcal{E}}\leqq\big{\|}E\big{\|}_{\mathcal{E}}\big{\|}F\big{\|}_{\mathcal{E}}$

Since $\|\mathcal{R}_{1}\|_{\mathcal{B}(\mathcal{E})}=1$ we have,
$\|\mathcal{R}_{E}(1)\|_{\mathcal{E}}=\|E\|_{\mathcal{E}}\leqq\|\mathcal{R}_{E}\|_{\mathcal{B}(\mathcal{E})}\leqq\|\mathcal{R}_{1}\|_{\mathcal{B}(\mathcal{E})}\>\|E\|_{\mathcal{E}}=\|E\|_{\mathcal{E}}.$
Similarly for $\mathcal{L}$. Then we must have

$\big{\|}\mathcal{R}_{E}\big{\|}_{\mathcal{B}(\mathcal{E})}=\big{\|}E\big{\|}_{\mathcal{E}}=\big{\|}\mathcal{L}_{E}\big{\|}_{\mathcal{E}}.$

Hence the embedding,

$\mathcal{R}:\mathcal{E}\rightarrowtail\mathcal{R}(\mathcal{E})$

is an isometric isomorphism to its image and $\mathcal{R}({\mathcal{E}})$ is a
Banach algebra. The space of states acts as continuous linear functionals on
$\mathcal{R}(\mathcal{E})$, given by the map
$\mathcal{R}_{E}\mapsto\big{\langle}\omega|E\big{\rangle}_{\mathcal{E}},$ with

$|\omega(\mathcal{R}_{E})|=|\langle\omega|E\rangle_{\mathcal{E}}|\leqq\|\mathcal{R}_{E}\|_{\mathcal{B}(\mathcal{E})}$

hence the elements of $\mathcal{A}_{*}$ respect the topological properties of
$\mathcal{R}(\mathcal{E})$.

We may now note what it means to re-perform an experiment or attempt to
measure the same outcome. For an outcome $R$, the preparation procedure
concatenated by a measuring instrument corresponding to the effect
$\mu(\cdot|R)$ is itself a preparation procedure. To such preparation
procedure, an application of the same measuring instrument again should not
change the ensemble the procedure produces, since the the belief about outcome
has already been verified. Since the concatenation of measurements corresponds
to the product structure of $\mathcal{R}(\mathcal{E})$, it must be the case
that whenever $E$ in $\mathcal{R}(\mathcal{E})$ corresponds to an experimental
outcome it must correspond to a projection element

$E^{2}=E.$

The effects of measuring outcomes correspond to orthogonal projections in
$\mathcal{R}(\mathcal{E})$.

## Complex Structure

As soon as $\mathcal{R}(\mathcal{E})$ possesses an algebra structure, we can
make sense of polynomials in $\mathcal{R}(\mathcal{E})$. Suppose a measuring
instrument consists of a collection of mutually compatible instruments, that
is, the measurements performed by such instruments do not affect the outcomes
of the other instruments. Suppose the instrument undergoes a change if it
undergoes a change for each of its component instruments or sub-instruments,
then the probability of outcome for such an instrument should correspond to
product of probabilities of its components.

Let $E$ be the effect corresponding to such an instrument and $E_{i}$ be the
effects corresponding to its components. Then for any ensemble $\omega$ we
must have,

$\big{\langle}\omega|E\big{\rangle}_{\mathcal{E}}=\prod_{i\in\mathcal{I}}\>\langle\omega|E_{i}\rangle_{\mathcal{E}}.$

Since each effect corresponds to an equivalence class of measuring
instruments, we may expect there to exist some measuring instrument which is
the collection of compatible measuring instruments which cannot be further
divided. This allows us to postulate that _it is always possible to decompose
a measuring instrument into its maximally divided collection of sub-
instruments_. We assume that it is always possible to decompose a measuring
instrument as in terms of its component sub-instruments. To make sense of this
we need structures that enable us to factorise polynomials in
$\mathcal{R}(\mathcal{E})$. By the fundamental theorem of algebra, we can
always factorise polynomials in the field of complex numbers. Hence we may
assume the underlying field to be that of complex numbers

$\mathbb{K}\equiv\mathbb{C}.$

Therefore we assume the pairing to be a real bilinear map to the field of
complex numbers

$\big{\langle}\cdot|\cdot\big{\rangle}_{\mathcal{R}(\mathcal{E})}:\mathcal{A}_{*}\times\mathcal{R}(\mathcal{E})\to\mathbb{C}.$

For mathematical convenience, we will transfer this complex structure to the
vector spaces $\mathcal{A}_{*}$ and $\mathcal{R}(\mathcal{E})$ themselves, and
assume that $\mathcal{A}_{*}$ and $\mathcal{R}(\mathcal{E})$ are complex
vector spaces. We require the scaling of both effects and ensembles by a unit
length complex number to give the same pairing as the unscaled pairing. This
requirement along with real linearity in each argument forces us to assume the
pairing is sesquilinear. We now have access to the theory of complex numbers,
and complex analysis.

Since the set of quaternions is not an algebra over $\mathbb{C}$, they do not
intrinsically allow for factorisation as required in the discussion above. On
the other hand, the structures in quaternions are not _required_ by the
postulates so far, and hence quaternions cannot be part of the universal
mathematical theory for these postulates. We may therefore rule out quantum
mechanics in the field of quaternions.

##  $W^{*}$-Algebra Structure

We now consider operations on the space of effects that are physically
meaningful. Suppose we perform a rearrangement or readjustments of all the
measuring instruments. Then we can consider all the measuring instruments
which have been turned to non-functioning instruments, these instruments never
undergo any changes, and the re-arrangement or re-adjustment of instruments
acts as a switch-off for those instruments. Assuming such manipulations keep
the system within itself, it must keep the space of effects invariant. However
the only way to find out what changes we have made to the system is by
performing experiments on the system. Since the space of states is separating
for the space of effects, any manipulation we do to the space of effects are
_seen_ by the space of states. Hence we consider the equivalence class of such
manipulations on the space of effects, which affect the pairing
$\langle\cdot|\cdot\rangle_{\mathcal{R}(\mathcal{E})}$ of
$\mathcal{R}(\mathcal{E})$ with $\mathcal{A}_{*}$ in the same way. Each such
equivalence class of manipulation of the system corresponds to a linear map,

$T:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E}).$

The manipulation $T$ of the system can also be thought of as a manipulation of
the preparation procedures. We can also consider the equivalence class of
manipulations of the system giving rise to the same change in preparation
procedures, and for every ensemble $\omega$ this can be described in terms of
operational statistics by the map

$R_{T}\omega:E\mapsto\big{\langle}\omega|TE\big{\rangle}.$

This map of ensembles can be extended to $\mathcal{A}_{*}$ by linearity, such
that for every preparation procedure $\rho$ and $\sigma$ and scalars $\alpha$
and $\beta$,
$\alpha\mu(\rho|\cdot)+\beta\mu(\sigma|\cdot)\mapsto\alpha\mu\big{(}T(\cdot)|\rho\big{)}+\beta\mu\big{(}T(\cdot)|\sigma\big{)}.$
Hence to each physically meaningful manipulation of the system there must
exist a bounded linear map,

$R_{T}:\mathcal{A}_{*}\to\mathcal{A}_{*},$

This map on $\mathcal{A}_{*}$ gives us a map,

$T^{\dagger}:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E}).$

It is determined by the pairing
$\langle\cdot|\cdot\rangle_{\mathcal{R}(\mathcal{E})}$ of
$\mathcal{R}(\mathcal{E})$ and $\mathcal{A}_{*}$ by,

$\big{\langle}T^{\dagger}\omega|E\big{\rangle}_{\mathcal{R}(\mathcal{E})}=\big{\langle}R_{T}\omega|E\big{\rangle}_{\mathcal{R}(\mathcal{E})}$

Note that $R_{T}$ can also be thought of in terms of the left-action
$\mathcal{L}$, since it corresponds to the transformation by $T$ of effects
which can be pre-composed with effects. Hence the $\dagger$-operation would
have the effect of reversing the order of taking product, when viewed in terms
of ensembles. This can be described more directly in terms of the pairing
$\langle\cdot|\cdot\rangle_{\mathcal{R}(\mathcal{E})}$.

By the sesquilinearity of the pairing
$\langle\cdot|\cdot\rangle_{\mathcal{R}(\mathcal{E})}$ it follows for all
physically relevant operators $T,S$ on the space $\mathcal{R}(\mathcal{E})$
and for all scalars $\lambda$, and $\mu$ that,

$\displaystyle({\lambda T+\mu S})^{\dagger}$
$\displaystyle=\overline{\lambda}T^{\dagger}+\overline{\mu}S^{\dagger},$
$\displaystyle(TS)^{\dagger}$ $\displaystyle=S^{\dagger}T^{\dagger}.$

The operation $\dagger:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E})$
seems to be completely algebraic in nature, however it also possesses
interesting topological properties, since the topologies on the space of
effects and state space operationally correspond to the physical notion of
accuracy. In order to describe the topological properties of the operation
$\dagger$, we now describe the notion of complementary effect and its relation
to the notion of topological complementation on Banach spaces.

Suppose we have a measuring instrument which can measure an outcome $R$, the
change occuring in the instrument corresponds to the effect for $R$. If the
measuring instrument does not undergo any change, then we can infer that the
outcome $R$ had not occurred. Since the outcome $R$ not occuring is also an
outcome by itself, denoted by $\neg R$, and the same measuring instrument also
be used for measuring $\neg R$, the instrument not undergoing a change
corresponds to the effect for $\neg R$. Hence we must have an operation on the
space of effects which assigns to every effect $\mu(\cdot|R)$ corresponding to
the outcome $R$, the effect $\mu(\cdot|{\neg R})$ corresponding to the outcome
$\neg R$. We call $\mu(\cdot|{\neg R})$ the complement of $\mu(\cdot|R)$.
Generally, we denote the complement of $E$ by $E^{\perp}$. We obtain a mapping
on the space of effects. This notion of complement of effects is closely
related to the notion of topological complementation of closed subspaces on
Banach spaces.

The Banach space structure on $\mathcal{R}(\mathcal{E})$ allows us to make
sense of closed subspaces of $\mathcal{R}(\mathcal{E})$, with respect to the
norm $\|\cdot\|_{\mathcal{B}(\mathcal{E})}$ inherited from
$\mathcal{B}(\mathcal{E})$. Let $\mathcal{M}$ be a closed subspace of
$\mathcal{R}(\mathcal{E})$, we say that $\mathcal{M}$ is topologically
complemented if there exists a closed subspace $\mathcal{M}^{\perp}$ of
$\mathcal{R}(\mathcal{E})$ such that

$\mathcal{M}\oplus\mathcal{M}^{\perp}\equiv\mathcal{R}(\mathcal{E}).$

In such a case every vector $A$ in $\mathcal{R}(\mathcal{E})$ can be uniquely
decomposed as a direct sum of vectors
$A_{\mathcal{M}}+A_{\mathcal{M}^{\perp}}$ where $A_{\mathcal{M}}$ is in
$\mathcal{M}$ and $A_{\mathcal{M}^{\perp}}$ is in $\mathcal{M}^{\perp}$. Hence
we can consider the projection map on $\mathcal{R}(\mathcal{E})$,

$E_{\mathcal{M}}:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E})$

which sends each $A$ in $\mathcal{R}(\mathcal{E})$ to the vector
$A_{\mathcal{M}}$ in $\mathcal{M}$. Since we do not have to decompose
$A_{\mathcal{M}}$ any further we must have
$E_{\mathcal{M}}=E_{\mathcal{M}}^{2}$.

The closed graph theorem suggests that the continuity properties of operators
on Banach spaces and the topology of the product spaces are closely related,
and says that an operator is continuous if and only if its graph is closed.
Since both $\mathcal{R}(\mathcal{E})$ and $\mathcal{M}$ are closed, it follows
that $E_{\mathcal{M}}$ has a closed graph, and hence the projection,

$E_{\mathcal{M}}:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E}),$

onto the closed complemented subspace $\mathcal{M}$ of
$\mathcal{R}(\mathcal{E})$ is a continuous operator on
$\mathcal{R}(\mathcal{E})$. Conversely, every continuous projection
$E_{\mathcal{M}}$, corresponds to a unique closed complemented subspace
$\mathcal{M}$. Since the space of effects is contained in
$\mathcal{R}(\mathcal{E})$ and since each effect correspond to a continuous
projection operator, it follows that every effect corresponds to a unique
closed complemented subspace of $\mathcal{R}(\mathcal{E})$, hence the notation
for the complement effect $E^{\perp}$ is closely related to the notion of
topological complementation on $\mathcal{R}(\mathcal{E})$.

Every laboratory can be thought of as a subsystem of the universe, and can
equivalently be described in terms of the preparation procedures it can
prepare, or by all the experiments which we can perform in the laboratory. By
switching off the laboratory, we switch off all the experiments we could
perform in the laboratory, or equivalently switch of all the preparation
procedures in the laboratory. The laboratory itself can be thought of as a
measuring instrument which is compatible with all the experiments it can
perform, and for the laboratory itself, this must correspond to the identity
element. Hence to each laboratory, there must exist a projection element,
which commutes with every experiment that can be performed in the laboratory.
To each preparation procedure in the laboratory, this projection element
corresponds to the switching on of the procedure. For each preparation
procedure, we have a collection of laboratories in which the preparation
procedure can exist. For example, a preparation procedure in a laboratory, is
also a preparation procedure in a bigger laboratory which contains the smaller
laboratory. The projection element corresponding to the smallest laboratory to
which the preparation procedure can belong to, will be called the support of
the preparation procedure. Hence we obtain a correspondence between projection
elements and collection of preparation procedures for which the projection
corresponds to the minimal laboratory. If $\mu(\rho|\cdot)$ is the ensemble
corresponding to some preparation procedure $\rho$, then we denote by
$E_{\rho}$ the projection element as described above. We must require that

$\big{\|}\mu\big{(}\rho\>|\cdot\big{)}\big{\|}_{\mathcal{A}_{*}}=\big{\|}\mu\big{(}\cdot|E_{\rho}\big{)}\big{\|}_{\mathcal{E}}.$

By linearity, the operator $(T^{\dagger})^{\dagger}$ is determined by the
pairing $\mu(\rho|(T^{\dagger})^{\dagger}(E))=\mu(R_{T}\rho|E)=\mu(\rho|TE)$
for all effects $E$ belonging to the support of $\mu_{\rho}$, and hence also
for $E_{\rho}^{\perp}$. By the uniqueness of the pairing of closed
complemented subspaces with its complement, and since ensembles and effects
separate each other, we must require that

$(T^{\dagger})^{\dagger}=T.$

For all physically relevant transformations $T$, the mapping $\dagger$ acts as
an involution. The uniqueness of the topological complementation ensures that
the operation $\dagger$ is an involution.

The space of effects $\mathcal{R}(\mathcal{E})$ is a subset of this
collection, and hence we can consider this space of physically relevant
operators on the space of effects itself as the object of study. Abusing
notation we denote this algebra of operators by $\mathcal{R}(\mathcal{E})$.
From the discussion so far, we expect $\mathcal{R}(\mathcal{E})$ to be a
Banach algebra equipped with an involution

$\dagger:\mathcal{R}(\mathcal{E})\to\mathcal{R}(\mathcal{E}).$

Such Banach algebras are called involutive Banach algebras, or
Banach∗-algebras.

However ensembles are themselves closely related to effects in an opposite way
via the notion of support. Hence the composite manipulation of the system acts
in an _independent_ way on the space of effects and the state space. From the
discussion about norms on $\mathcal{R}(\mathcal{E})$ and $\mathcal{A}_{*}$, we
know that the norms are closely related to the notion of accuracy of measuring
instruments and preparation procedures. Since the manipulations introduced on
the space of effects and state space are opposite of each other they are
independent, and we should expect the accuracies of effects and ensembles to
be multiplicative. The relation between support properties of ensembles and
effects captures the complementary nature of such a composite manipulation
$T^{\dagger}T$ of the system.

We now make this precise, via topological complementations and the expected
relationship between subspaces of the state space and ideals of the space of
effects. The norm of $T^{\dagger}T$ is given by

$\big{\|}T^{\dagger}T\big{\|}=\sup_{E\in\mathfrak{E}\atop\omega\in\mathfrak{S}}\Big{[}\big{|}\big{\langle}R_{T}\omega|TE\big{\rangle}_{\mathcal{R}(\mathcal{E})}\big{|}\Big{]}.$

We denote the space ${\mathcal{B}(\mathcal{R}(\mathcal{E}))}$ of bounded
operators on $\mathcal{R}(\mathcal{E})$. We must have,

$\displaystyle\big{\|}T^{\dagger}T\big{\|}=\sup_{E\in\mathfrak{E}\atop\omega\in\mathfrak{S}}\Big{[}\big{|}\big{\langle}R_{T}\omega|TE\big{\rangle}_{\mathcal{R}(\mathcal{E})}\big{|}\Big{]}=\sup_{E\in\mathfrak{E}}\Big{[}\big{\|}TE\big{\|}\Big{]}\sup_{\omega\in\mathfrak{S}}\Big{[}\big{\|}R_{T}\omega\big{\|}\Big{]}=\big{\|}T\big{\|}^{2}.$

The reason for the equality of products of supremums is due to the fact that
for every $\omega$ we can split the state space as direct sum of complemented
subspaces as
$\mathcal{A}_{*}=(E_{R_{T}\omega})\mathcal{A}_{*}\oplus({E_{R_{T}\omega}}^{\perp})\mathcal{A}_{*}$
and the pairing vanishes if $TE$ does not belong to the support of
$R_{T}\omega$, we can assume that the supremum is attained only when $TE$
belongs to the ideal $({E_{R_{T}\omega}}^{\perp})\mathcal{E}$. Hence we have,

$\big{\|}T^{\dagger}T\big{\|}=\big{\|}T\big{\|}^{2}.$

Banach∗-algebras satisfying this identity are called $C^{*}$-algebras. The
following composite map gives us an embedding of $\mathfrak{E}$ inside
${\mathcal{B}(\mathcal{R}(\mathcal{E}))}$,

$R\mapsto\mu(\cdot|R)\mapsto\mathcal{R}_{\mu(\cdot|R)}\mapsto\mathcal{R}_{\mathcal{R}_{\mu(\cdot|R)}}.$

Note that it is sufficient to consider only the bounded operators, and any
other unbounded yet meaningful manipulations can be described as maps on the
bounded operators. The algebra of observables $\mathcal{A}$ is the collection
of all elements of $\mathcal{B}(\mathcal{R}(\mathcal{E}))$ which are bounded
and correspond to physically meaningful manipulations of the system. We must
have a representation of measuring instruments as an embedding inside the
algebra of observables. Since any changes from such manipulations must be
noticed by the space of ensembles, it follows that $\mathcal{A}_{*}$ remains
separating for $\mathcal{A}$. Hence we must have a dual pairing,

$\big{\langle}\cdot|\cdot\big{\rangle}:\mathcal{A}_{*}\times\mathcal{A}\to\mathbb{C}.$

Such that,

$(\mathcal{A}_{*})^{*}\cong\mathcal{A}$

A $C^{*}$-algebra $\mathcal{A}$ which is a dual Banach space of
$\mathcal{A}_{*}$ is called a $W^{*}$-algebra, and in such a case the Banach
space $\mathcal{A}_{*}$ is called the predual of $\mathcal{A}$.

We may expect any operational theory to be represented by the pairing of a
$W^{*}$-algebra, and its predual. By Sakai’s predual theorem, see [10]-[12],
$W^{*}$-algebras are the same as von Neumann algebras on a Hilbert space.
Hence every quantum theory naturally requires the structures of Hilbert spaces
and bounded operators on Hilbert spaces.

### Philosophical Digression: Subjectivity vs Objectivity

If we view ensembles as a representation of an experimenter’s belief about
reality, the collection of effects allow the experimenter to verify his
beliefs about the universe. The results of experiments represent an updating
of the experimenter’s beliefs. If we define an observer as an equivalence
class of experimenters with the same set of beliefs, then we may view
operational theories _as the theory of observers and the observed._ The set of
ensembles corresponds to the set of all beliefs the observer _can_ hold, and
represents the observer’s _subjective reality_. Since the space of effects is
the only link between an experimenter and objective reality, we may identify
the space of effects with objective reality itself. From this point of view,
we may view operational theories as theories of duality between the subjective
and the objective.

On the other hand, each physical theory can itself be thought of as an
observation about how observers observe reality. If we expect there to exist
some physical theory that allows us to predict about the physical world to any
level of accuracy then reality can be thought of as a limit of physical
theories. Then _reality is a functor categorical object_ , corresponding to
the limit of physical theories in the convenient complete category of functors
from the category of physical theories to the category of sets. Where the
category of physical theories consists of physical theories as objects, and
relation between physical theories as morphisms. Physical theories themselves
act as _higher effects_ and physics itself acts as _higher reality_ , and
mathematics may itself thought of as the collection of _higher ensembles_.

### Acknowledgements

I extend my sincere thanks to my master’s advisor, Dr. Günther Hörmann, for
their invaluable guidance and support during my research. Their expertise and
encouragement have been crucial to the completion of this work.

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