## 36th CANADIAN OPERATOR SYMPOSIUM (COSy)

May 20-24, 2008

University of Toronto, Ontario, Canada

## Speaker Abstracts

*Unitary Groups As A Complete Invariant*

by

**Ahmed Al-Rawashdeh**, Jordan University of Science and Technology,

Coauthors: Thierry Giordano

In 1954, Dye proved that the unitary groups of von Neumann factors
not of type I2n determine the algebraic type of factors. Using Dye's
result, Broise showed that any isomorphism between the unitary groups
of two von Neumann factors not of type In is implemented by a linear
or a conjugate linear *-isomorphism between the factors. Using Dye's
approach, we prove that the unitary groups determine the algebraic
types for a large class of simple, unital C*-algebras such as the
tracial topological rank zero (TAF-algebras) whose K1 groups are
isomorphic and a large class of simple, unital purely infinite nuclear
C*-algebras. Indeed, If j is an isomorphism between the unitary
groups of such C*-algebras (as above, including the irrational rotation
algebras and the simple unital AF-algebras, the Cuntz algebras),
then it induces a bijection between the sets of projections which
preserves the orthogonality and the unitarily equivalence of projections,
afterwards this mapping induces an isomorphism between their ordered
K0-groups.

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*Ideal structure of NF algebras and
the UCT revisited*

**Bruce Blackadar**, University of Nevada, Reno

This talk will be in two parts. In the first, we will discuss
how to describe ideals in NF algebras and their relationship with
a generalized inductive system, and a possible approach to showing
that every stably finite separable nuclear C*-algebra is an NF algebra.
In the second part, we will discuss the Universal Coefficient Theorem
and a possible obstruction to its general validity.

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**A Dynamical Core for Topological Quivers**

**Berndt Brenken**

A topological quiver E may be viewed as a directed graph with edge
and vertex spaces given by topological spaces. For a given quiver
we construct and abstractly characterize a subquiver yielding the
iterative dynamical core of the original quiver. The associated
Cuntz-Pimsner C*-algebra is a quotient of the Cuntz-Pimsner C*-algebra
of E.

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**The Cuntz Semigroup**

**Nate Brown**, Penn State

I'll discuss some theory and applications of the Cuntz semigroup.

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*Normalizers of subfactors*

Jan Cameron, Texas A&M University

It is sometimes of interest to describe the normalizer of a subalgebra
B of a II_1 factor M, that is, to describe the group N_M(B) of unitaries
in M that fix B under conjugation and the von Neumann algebra that
the group generates. We present structure results for both the group
N_M(B) and the associated von Neumann algebra for the case in which
B is a subfactor. Inclusions arising from the crossed product, group
von Neumann algebra, and tensor product constructions will also
be discussed.

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*Non-commutative Magic*

Man-Duen Choi, Department of Mathematics, University of
Toronto

Suddenly, it comes to the era of quantum computers, where non-commutative
matrix analysis will play a central role in many concrete applications.
Herein, I will show the magic of real computations in connection
with obvious phenomena of non-commutative probability and non-commutative
geometry.

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*Equivalence between two invariants
for C*-algebras*

Alin Ciuperca*, *University of Toronto

We show that, for a C*-algebra of stable rank one, two isomorphism
invariants, the Cuntz semigroup and the Thomsen semigroup, contain
the same information. Several applications of this result will be
discussed.

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*One-parameter continuous fields of
AF-algebras*

Marius Dadarlat, Purdue University

Coauthors: George A. Elliott and Zhuang Niu

We show that one-parameter separable unital continuous fields of
AF-algebras are classified by their ordered K-theory sheaves. We
prove Effros-Handelman-Shen type theorems for separable unital one-parameter
continuous fields of AF-algebras and Kirchberg algebras.

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**Topological stable rank of Banach
algebras**

**Ken Davidson**, University of Waterloo

Coauthors: You Qing Ji, Rupert Levene, Laurent Marcoux, Heydar Radjavi

25 years ago, Rieffel introduced an algebraic invariant for Banach
algebras called topological stable rank which generalized the notion
of dimension to the non-commutative setting. The topological stable
rank has a left and right version, which coincide for C*-algebras
and commutative algebras. Moreover, tsr is a Banach algebra variant
of the purely algebraic invariant of Bass stable rank for rings-and
the left and right versions of Bass stable rank are always equal.
So Rieffel asked whether they are always equal? We have calculated
the left and right topological stable ranks for the class of nest
algebras, and can answer Rieffel's question negatively.

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*Nonseparable UHF algebras*

**Ilijas Farah, **York University

Coauthors: Takeshi Katsura

Separable UHF algebras were classified in the 1960s by Glimm and
Dixmier. Dixmier asked whether three standard definitions of UHF
algebras are equivalent in the nonseparable unital case. We answer
this question and prove some results (both positive and negative)
about the structure of general nonseparable C* algebras.

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**Injective envelopes of continuous
trace C*-algebras**

**Doug Farenick**, University of Regina

Coauthors: Martin Argerami and Pedro Massey

If A is a postliminal (type I) C*-algebra, then its injective envelope
I(A) is a type I AW*-algebra. The ideal J of I(A) generated by the
abelian projections of I(A) is a liminal C*-algebra with Hausdorff
spectrum, and its multiplier algebra M(J) is I(A). Because J can
be represented as a continuous C*-bundle, the multiplier algebra
of M(J)=I(A) is a C*-algebra of bounded, strictly continuous operator
fields on the spectrum of J. How is the spectrum of J determined
from A? In particular, what is the relationship between the spectra
of A and J if A is assumed to be a continuous C*-bundle over a locally
compact Hausdorff space? This talk will address such questions,
with the overall aim of finding an explicit description of the injective
envelopes and the local multiplier algebras of continuous trace
C*-algebras A.

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*The asymptotic flow of an E0-semigroup*

Remus Floricel, University of Regina

The asymptotic flow of an $E_0$-semigroup $\rho=\{\rho_t\,|\,t\geq
0\}$ acting on a type $\rm{I}_{\infty}$ factor $M$ is the $W^*$-dynamical
system $(M_\infty, \rho\mid_{M_\infty})$, where $M_\infty$ is the
tail algebra $\bigcap_{t>0}\rho_t(M)$. We show that for any $E_0$-semigroup
$\rho$, and $s>0$, there exists a cocycle perturbation of $\rho$
whose asymptotic flow is a $s$-periodic type $\rm{I}_{\infty}$ $W^*$-dynamical
system.

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*C-Orbit Reflexivity of Hilbert Space
Operators*

Ileana Ionascu*,* Philadelphia University

The talk will present a characterization of C-Orbit reflexivity
on finite-dimensional spaces.

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** **

**Invariant subspaces of non-associative
algebras of compact operators.**

**Matthew Kennedy**, University of Waterloo

Coauthors: Victor Shulman, Yuri Turovskii

Several classical results imply the (simultaneous) triangularizability
of certain non-associative algebras of nilpotent matrices. These
include Engel’s theorem for Lie algebras, and Jacobson’s
generalization which applies to Jordan algebras. This talk is about
recent extensions of these results to the setting of compact operators
in infinite dimensions, which can be viewed as an application of
a more general result about the existence of invariant subspaces
for certain subgraded Lie algebras of compact operators.

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*Braidability*

**Claus Koestler**, University of Illinois at Urbana-Champaign

Coauthors: Rolf Gohm

We introduce `braidability' as a symmetry which is intermediate
to the distributional symmetries `exchangeability' and `spreadability'
of noncommutative infinite random sequences. This endows the braid
groups Bn with a new intrinsic (quantum) probabilistic interpretation.
We show as an application that certain unitary representations of
the braid group B8 are accompanied by Jones-Temperley-Lieb algebras.

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*Complementarity in Quantum Cryptography
and Error Correction*

David Kribs, University of Guelph

Coauthors: Dennis Kretschmann and Robert Spekkens

In this talk, I'll outline recent work that shows how two basic
notions in quantum cryptography and quantum error correction are
complementary to each other. Error-correcting codes for quantum
channels are the key vehicles used to avoid noise such as decoherence
in quantum computing. Private codes for quantum channels play a
central role in quantum communication and cryptography. It turns
out that a code is private for a channel precisely when it is correctable
for a complementary channel, and there is a straightforward algebraic
recipe that allows one to move between the two perspectives. Moreover,
an approximate version of the relationship can be proved in terms
of diamond (or completely bounded) norms for channels.

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*Z-stability, purely infinite corona,
and skeletons *

Dan Kucerovsky, University of New Brunswick, Fredericton

Purely infinite corona can be viewed as a strong form of the corona
factorization property. We study both of these properties in detail
in the case of real rank zero C*-algebras, and give preliminary
results on purely infinite corona for Z-stable C*-algebras, not
necessarily of real rank zero.

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*The Hecke algebra of Bost-Connes revisited
*

**Magnus Landstad, **NTNU, Trondheim, Norway

I shall look at some (but not all) developments after the introduction
of the Hecke algebra of Bost and Connes. A general Hecke pair (G,
H) where G is a group and H a subgroup can more easily described
via its Schlichting completion (G*, H*) with H* a compact open subgroup
of G* (due to Tzanev).

To a Hecke algebra there is also a Banach *-algebra and a C*-algebra
and for the representation theory of these 3 algebras both Fells
and Rieffels version of Morita equivalence is needed. Some old
results about Banach *-algebras reappear.

The BC Hecke algebra ax+b-group has two nice properties we shall
study separately:

1) The semigroup S={s| ad(s) maps H into H} gives an ordering of
G.

2) G is a semi direct product QN, where H is a subgroup of the normal
subgroup N.

Time permitting, we may also look at

3) Generalised Hecke algebras, here it turns out that there is a
different Schlichting completion of (G, H).

4) Cuntz ax+b-semigroup and why does it contain the BC Hecke algebra?

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*Projectivity of L^p(VN(G)) as a left A(G)-module*

**Hun Hee Lee**, University of Waterloo

Coauthors: Brian Forrest and Ebrahim Samei

For any locally compact group G we can understand L^p(G) as a L^1(G)(the
convolution algebra)-module under the convolution product. Dales
and Polyakov (2004) proved that, for finite p which is greater than
1, L^p(G) is projective as a left L^1(G)-module if and only if G
is compact. In this talk we focus the dual situation, namely a natural
A(G)(the Fourier algebra)-module structure on L^p(VN(G)).We will
show that, for finite p which is greater than 1, L^p(VN(G)) is a
projective left operator A(G)-module when G is discrete and amenable.Conversely,
we can show that, for finite p which is greater or equal to 2, L^p(VN(G))
is not projective when G is non-discrete group with approximation
property. Moreover, the above module structure can be defined similarly
in the case of Kac algebras and locally compact quantum groups,and
some results about projectivity still hold.

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*Realizing irreducuble semigroups and
real algebras of compact operators*

**Mitja Mastnak**, University of Waterloo

Coauthors: J Bernik, H. Radjavi

Let B(X) be the algebra of bounded operators on a complex Banach
space X. Viewing B(X) as an algebra over R, we study the structure
of those irreducible subalgebras which contain nonzero compact operators.
In particular, irreducible algebras of trace-class operators with
real trace are characterized. This yields an extension of Brauer-type
results on matrices to operators in infinite dimensions, answering
the question: is an irreducible semigroup of compact operators with
real spectra realizable, i.e., simultaneously similar to a semigroup
whose matrices are real?

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*The Maximal C*-Algebra of Quotients
as an Operator Bimodule*

**Martin Mathieu**, Queen's University Belfast

Coauthors: Pere Ara (Barcelona), Eduard Ortega (Odense)

We establish a description of the maximal C*-algebra of quotients
of a unital C*-algebra A as a direct limit of spaces of completely
bounded bimodule homomorphisms from certain operator submodules
of the Haagerup tensor product A?h A labelled by the essential closed
right ideals of A into A. In addition the invariance of the construction
of the maximal C*-algebra of quotients under Morita equivalence
is proved.

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*Correlations of Eigenvalues of Random
Matrices*

Jamie Mingo

Given an $N \times N$ self-adjoint random matrix we get a random
set of $N$ real eigenvalues. For many random matrix models we know
the limiting eigenvalue distribution (as $N$ tends to infinity)
and the limiting distribution is not random. However the eigenvalues
themselves are not independent and so what we see for finite $N$
is not $N$ independent samples from a distribution close to the
limiting distribution. Indeed there is often repulsion between the
eigenvalues. The correlations are quantified by $k$-point functions.
We will examine the limiting correlations for small values of $k$
and some standard matrix ensembles.

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*Nuclearity and weak uniqueness*

**Ping Wong Ng**, University of Louisiana

Coauthors: A. Ciuperca and Z. Niu

We characterize nuclearity using a weak uniqueness theorem, where
the invariant is the Cuntz semigroup.

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*Some properties of truncated Haar unitary
random matrices*

Jonathan Novak, Queen’s University

The spectrum of a Haar unitary random matrix is a determinantal
point process on the unit circle, while the spectra of its submatrices
are determinantal processes in the unit disc. We derive formulas
for the moments of the norm of the trace of such submatrices. This
leads to connections between random matrix theory, the enumeration
of lattice walks confined to Weyl chambers, and Toeplitz determinants
of Bessel functions.

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*Properties of Generalized Bunce-Deddens
algebras*

**Stefanos Orfanos**, Purdue university

We define a version of generalized Bunce-Deddens algebras as certain
crossed products by discrete amenable residually finite groups,
and we describe some of their properties.

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**Connected MASAs in UHF algebras **

**Chris Phillips,** University of Oregon, Eugene

Coauthors: Simon Wassermann

It has been an open question for some time whether there is a maximal
abelian subalgebra in a UHF algebra which is isomorphic to C (X)
for a connected space X. In this talk, we describe a method for
producing uncountably many mutually nonconjugate maximal abelian
subalgebras in the CAR algebra (the 2^{\infty} UHF algebra), each
isomorphic to C ([0, 1]).

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*An Index Theory for Certain Gauge
Invariant KMS States on C*- Algebras*

John Phillips, University of Victoria

We present, by examples, an index theory appropriate to algebras
without trace. In particular, our examples include the Cuntz algebras
and a larger class of unital separable C*-algebras that generate
all injective III factors for 0 < < 1: These algebras
are denoted by O and include the Cuntz algebras: O 1=n = O n :
Our main result is an index theorem (formulated in terms of spectral
using a twisted cyclic cocycle where the twisting comes from the
modular automorphism group for a natural gauge (circle action) invariant
KMS state. We introduce a modied K 1 -group for these algebras
that we can pair with this twisted cocycle. As a corollary we obtain
a noncommutative geometry interpretation for Araki's notion of relative
entropy in these examples. This is joint work with Alan Carey and
Adam Rennie.

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*On local-to-global properties of semigroups
of operators*

**Heydar Radjavi**, University of Waterloo

Let S be a semigroup of operators with no common, nontrivial, invariant
subspaces. What can be said about S if there exists a nonzero linear
functional f (on all operators) whose restriction to S takes only
real values or, more restrictively, only positive values? Even when
confined to compact operators, not all the questions have trivial
answers.

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*The cones of lower semicontinuous traces
and quasitraces of a C*-algebra*

Leonel Robert, University of Toronto

Coauthors: George A. Elliott and Luis Santiago

In the classification of nonsimple C*-algebras it is necessary
to consider lower semicontinuous traces that are not necessarily
bounded or densely finite. All such traces form a (noncancellative)
topological cone. I will present some basic properties of this cone
that are relevant to classification questions. I will present applications
of these results to the computation of the Cuntz semigroup of the
following classes of C*-algebras: (1) C*-algebras with no simple
subquotients that absorb the Jiang-Su algebra, (2) simple C*-algebras
that absorb the Jiang-Su algebra (this computation was obtained
previously by Brown, Perera, and Toms, for the case of nuclear,
unital, stable rank 1 C*-algebras).

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*Uniform continuity over locally compact
quantum groups*

**Volker Runde,** University of Alberta

We define, for a locally compact quantum group G in the sense of
Kustermans-Vaes, the space of LUC(G) of left uniformly continuous
elements in L8(G). This definition covers both the usual left uniformly
continuous functions on a locally compact group and Granirer's uniformly
continuous functionals on the Fourier algebra. We show that LUC(G)
is an operator system containing the C*-algebra C0(G) and contained
in its multiplier algebra M(C0(G)). We use this to partially answer
an open problem by Bedos-Tuset: if G is co-amenable, then the existence
of a left invariant mean on M(C0(G)) is sufficient for G to be amenable.
Furthermore, we study the space WAP(G) of weakly almost periodic
elements of L8(G): it is a closed operator system in L8(G) containing
C0(G) and - for co-amenable G - contained in LUC(G). Finally, we
show that - under certain conditions, which are always satisfied
if G is a group - the operator system LUC(G) is a C*-algebra.

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*Resolvents and distances between operators*

**Roland Speicher,** Queen's University, Kingston

I will report on some ideas to measure the distance between operators
(respectively their distributions) in terms of resolvents. Estimating
distances to semicircular elements will feature prominently.

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**Hilbert modules over C*-dynamical systems**

**Andrew Toms** York University

The Cuntz semigroup has recently received much attention as an
invariant for nuclear separable C*-algebras.One consequence of this
study has been the discovery that isomorphism classes of Hilbert
modules over such algebras can, in the setting of stable rank one,
be classified by the Cuntz semigroup. When the natural partial order
on the Cuntz semigroup is determined by states, this classification
can be realised in terms of K-theory and traces. We show that the
order on the Cuntz semigroup is so determined for the C*-algebras
associated to minimal diffeomorphisms.

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**An anti-classification Theorem for
von Neumann factors**

by

**Asger Tornquist** University of Toronto

Coauthors: Roman Sasyk

We show that there is no reasonable way to classify von Neumann
factors on a separable Hilbert space by an assignment of invariants
which are "countable structures", e.g. countable groups
or graphs, up to isomorphism. We also show that the isomorphism
relation of factors is complete analytic. In particular, it is not
Borel.

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*Proper actions on $C^*$-algebras*

Dana Williams, Dartmouth College

Coauthors: Astrid an Huef (University of New South Wales), Iain
Raeburn (University of Wollongong)

In 1990, Rieffel formulated the notion of a proper $C^{*}$-dynamical
system $(A,G,\alpha)$. Under reasonable hypotheses, the corresponding
reduced crossed product $A\times_{\alpha,r}G$ is Morita equivalent
to a ``generalized fixed point algebra'' $A^{\alpha}$ in the multiplier
algebra $M(A)$. In this talk, I will discuss a number of results
concerning proper actions and their generalized fixed point algebras.
These results are most efficiently stated by showing that our constructions
are functorial. This is work in progress with Astrid an Huef and
Iain Raeburn.

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*On a class of Hilbert C*-manifolds*

by

**Wend Werner**, Westfälische Wilhelms-Universität,
Mathematisches Institut, Einsteinstraße 62, 48149 Münster,
Germany

Denote by U the open unit ball of a C*-algebra. U is a symmetric
space, where the transitively operating Lie group consists of all
biholomorphic automorphisms of U.

This talk has two objectives, an explicit calculation, for all
vector fields on U, of the invariant connection and, using results
previously obtained with D. Blecher, to characterize those invariant
cone fields that can be thought of as the result of some kind of
`quantization'. (In general relativity, such a structure is responsible
for the concept of causality.)

Both questions are related since the invariance of the cone fields
is intimately connected to the behavior of parallel transport along
geodesics.

Our results actually cover a much broader class of (infinite dimensional)
symmetric spaces. Important here is to use an invariant Hilbert
C*-structure on the fibers of the tangent bundle of U. We show that
the symmetric space we are dealing with can be defined in terms
of the automorphism group of this structure. For the underlying
invariant (operator space) Finsler structure, the analogous result
holds. It also turns out that the connection we are dealing with
relates to the Hilbert C*-structure in quite the same way as the
Levi-Civita connection does to its Riemannian metric.

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**A short survey of Burnside type theorems**

by

**Bamdad R. Yahaghi**, School of Mathematics, IPM, Tehran, Iran

A version of a celebrated theorem of Burnside asserts that Mn(F)
is the only irreducible subalgebra of Mn(F) provided that the field
F is algebraically closed. In other words, Burnside's theorem characterizes
all irreducible subalgebras of Mn(F) whenever F is algebraically
closed. In view of this, by a Burnside type theorem for certain
irreducible subalgebras of matrices, we mean a result which characterizes
such subalgebras. In this talk, we present a simple proof of Burnside's
theorem. We also present Burnside type theorems for irreducible
subalgebras of Mn(R), a result which is well known to the experts,
and for irreducible subalgebras of Mn(H), where H denotes the division
ring of quaternions. For a given n > 1, we characterize all fields
F for which Burnside's Theorem holds in Mn(F). If time permits,
letting K be a field and F a subfield of K which is k-closed for
all k dividing n with k > 1, we present a Burnside type theorem
for irreducible F-algebras of matrices in Mn(K) on which trace is
not identically zero. (For a k > 1, a field F is said to be k-closed
if every polynomial of degree k over F is reducible over F, e.g.,
R is k-closed for any odd integer greater than one.)

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**Representations of Higher Rank Graph Algebras**

by

**Dilian Yang**, University of Waterloo

Coauthors: Kenneth R. Davidson

We show that every irreducible atomic *-representation of a k-graph
is the minimal *-dilation of a group construction representation.
It follows that every atomic representation decomposes as a direct
sum or integral of such representations. We characterize periodicity
of a k-graph and identify a symmetry subgroup of Zk. If this has
rank s, then the graph C*-algebra is a tensor product of C(Ts) with
a simple C*-algebra.

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