# SCIENTIFIC PROGRAMS AND ACTIVITIES

July 28, 2014

## 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

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
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,

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

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 Fells and Rieffels 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

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 modi ed 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

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.