October 18, 2017

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

June 23-27, 2014
42nd Canadian Annual Symposium on Operator Algebras and Their Applications (COSy)

Location: June 23-24 & June 26-27, Fields Institute, 222 College St.
Location: June 25, Bahen Building, 40 St. George St. Room 1170


SPEAKER ABSTRACTS

Plenary Speakers


David Blecher (Houston)
'Interpolating' between Hilbert space operators, and real positivity for operator algebras

With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. As motivation note that the `completely' real positive maps on C*-algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C*-algebra it generates, and in particular what to we call noncommutative peak interpolation, and noncommutative peak sets. We report on the state of this theory (joint work with Read, ome in progress at the time of writing) and on the parts of it that generalize further to certain classes of Banach algebras (joint work with N. Ozawa).

Ken Davidson (Waterloo)
Semicrossed products over semigroups

If $P$ is a subsemigroup of a group $G$ that acts on a C*-algebra $A$ by $*$-endomorphisms, we construct the semicrossed product, which is a universal nonself-adjoint operator algebra with respect to some specified family of covariant representations. We seek to identify the C*-envelope, which is the smallest enveloping C*-algebra, as a crossed product of a related C*-algebra $B$ by an automorphic action of $G$. This is accomplished in a number of cases.
This is joint work with Adam Fuller (U. Nebraska) and Evgenios Kakariadis (Ben-Gurion U.).

Ilijas Farah (York)
A new tool for constructing nuclear C*-algebras

By combining old and new tools from model theory one can give unified proofs of various permanence properties for C*-algebras defined by approximation properties. These tools also provide new constructions of C*-algebras with prescribed properties. This is a report on work in progress with several coauthors, including B. Hart, M. Magidor, L. Robert and A. Tikuisis.

Farzad Fathizadeh (Caltech and Western )
Scalar Curvature and Gauss-Bonnet Theorem for Noncommutative Tori
Coauthors; Masoud Khalkhali

After the seminal work of Connes and Tretkoff on the Gauss-Bonnet theorem for the noncommutative two-torus, there have been significant developments in understanding the local differential geometry of these C*-algebras equipped with curved metrics. In this talk, I will review my joint works with M. Khalkhali in which we extend this result to general translation invariant conformal structures on noncommutative two-tori and compute the scalar curvature. Our final formula for the curvature matches precisely with the one computed independently by A. Connes and H. Moscovici. A purely noncommutative feature in these works is the appearance of the modular automorphism from Tomita-Takesaki theory in the computations and final formulas for the curvature. Time permitting I shall also try to indicate an extension of these results to curved noncommutative four tori.

Fereidoun Ghahramani (Manitoba)
An introduction to approximate amenability for operator algebraists

This talk is on basic material from theory of {\it approximate amenability} and its branches that R.J. Loy and I introduced in the year 2000 and have been developing since then, sometimes with other colleagues: Y. Choi, H.G. Dales, C.J. Read, E. Samei, and Y. Zhang. I have chosen the material for the talk so that it might be interesting for operator algebraists. Although some results to be presented are about operator algebras on Banach spaces, the talk, however, will end up with some open questions on operator algebras on Hilbert spaces.

Thierry Giordano (Ottawa)
Purely infinite partial crossed products

Partial actions of a discrete group on C*-algebras and their associated crossed products were introduced by Ruy Exel and Kevin McClanahan, and since then have been developed by many authors. In a recent work, Adam Sierakowski and I pursued the study of partial C*-dynamical systems and their associated crossed products. In this talk I will report on some of the results we obtained.

Guihua Gong (Puerto Rico)
TBA

Matthew Kennedy (Carleton)
Boundaries of reduced C*-algebras of discrete groups

For a discrete group G, we consider the minimal C*-algebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the C*-algebra of continuous functions on Furstenberg’s universal G-boundary. This operator-algebraic construction of the Furstenberg boundary turns out to have some interesting consequences. In particular, it leads to proof of a conjecture of Ozawa about nuclear embeddings of reduced C*-algebras of exact groups.

Masoud Khalkhali (Western)
Spectral Geometry of Curved Noncommutative Tori

While the algebraic topology of noncommutative tori, as reflected in their K-theory, cyclic cohomology and index theory, have been intensively studied in the past three decades, much less is known about their geometry.
Ideas of spectral geometry, even the very notion of Riemannian metric itself, can often be imported to noncommutative geometry thanks to Connes' notion of spectral triples. This gives an opening to geometric studies of noncommutative spaces endowed with a suitable notion of curved metric. In this talk I shall survey recent joint work with Farzad Fathizadeh, and the closely related work of Connes-Tretkoff and Connes-Moscovici, on Gauss-Bonnet theorem and the scalar curvature for curved
noncommutative tori $A_{\theta}^n $ for $ n=2, 4.$ In dimension 2 the local expression for curvature, as an element of the noncommutative torus, is computed by evaluating the value of the (analytic continuation of the) spectral zeta function $\zeta_a(s) = \text{Trace} (a |D|^{-s})$ at $s=0$ as a linear functional in $ a\in C^{\infty}({T}_{\theta}^2)$. In higher dimensions scalar curvature is related to the residue of the spectral zeta functions at its subleading pole.

Eberhard Kirchberg (HU Berlin)
`Central sequence algebras with and without characters

The ``corrected'' central sequence algebra F(A) of a separable C*-algebra A contains basic information about properties of A. An important question is: When F(A) is character-less? We report on two different aspects of this questions that we study in papers with Mikael Rordam and with Hiroshi Ando. E.g. we show that F(A) contains a unital C*-subalgebra that has B(H) as a unital quotient if A is anti-liminal. We give examples where F(A) has a character.
If F(A) has no character then A has the strong corona factorization property of Kucerovsky and Ng.
Additional assumptions on $A$ , like a rather strong 2-splitting property or divisibility and comparability properties, allow to show that
F(A) contains the the Jiang-Su algebra Z unitally as subalgebra (i.e., A absorbs Z tensorial) if F(A) has no character.

James Mingo (Queen’s)
Freeness and the Transpose

Over twenty years ago Voiculescu showed that independent and unitarily invariant random matrices are asymptotically free. Asymptotic freeness gives universal rules for computing (asymptotically) the eigenvalue distribution of sums and products of random matrices provided that one knows the individual eigenvalue distribution.

The requirement that the ensembles be unitarily invariant was lifted recently but the requirement that the ensembles be independent has always been necessary until a year ago when Mihai Popa and I showed that a matrix can be asymptotically free from its transpose. Recently we have show that this can pushed to the partial transposes of interest in quantum information theory.

Magdalena Musat (Copenhagen)
Factorizable completely positive maps and the Connes embedding problem

Zhuang Niu (Wyoming)
The C*-algebra of a minimal homeomorphism with zero mean dimension

Consider a minimal homeomorphism of a compact metrizable space, and assume that it has zero mean topological dimension (a dynamical version of topological covering dimension which was introduced by Lindenstrauss and Weiss). It is shown that the crossed product C*-algebra absorbs Jiang-Su algebra tensorially. This is a joint work with George A. Elliott.

N. Christopher Phillips (Oregon)
Examples of operator algebras on L^p spaces: Simplicity, uniqueness theorems, and amenability

We describe several classes of operator algebras on L^p spaces which, despite not having an adjoint, share some good properties of C*-algebras, and about which enough can be said to suggest that there may be a rich theory of such algebras. Our examples include:

L^p analogs of UHF and AF algebras.
L^p analogs of Cuntz algebras.
Full and reduced group L^p operator algebras.
L^p operator crossed products, particularly by free minimal actions
on compact metric spaces.
The L^p analog of the C*-algebra of the unilateral shift.

We consider simplicity, uniqueness results (analogous to those for the Cuntz C*-algebras), pure infiniteness, K-theory, amenability, p-nuclearity, and the relations between the structure of a group and L^p operator algebras built using it. A great many open problems remain. (There is a pdf file of some of them on the website of the 2014 GPOTS [Great Plains Operator Theory Seminar], at
http://www.math.ksu.edu/events/conference/gpots2014/LpOpAlgQuestions.pdf.)

This talk is related to my recent talk at GPOTS, but differs substantially. It has a different emphasis, and some of what I said in the earlier talk is already out of date. (There is a TeX version of my GPOTS talk on the conference website, at
http://www.math.ksu.edu/events/conference/gpots2014/Phillips_loaq.)


Mikael Pichot (McGill)
TBA

Ian Putnam (Victoria)
Smale spaces, their C*-algebras and a homology theory for them.

I will describe some hyperbolic topological dynamical systems called Smale spaces which were introduced by David Ruelle, giving a number of different examples including shifts of finite type. I will discuss the construction of various C*-algebras from these systems; in the case of shifts of finite type these include the Cuntz-Krieger algebras and their AF cores. I will also describe a homology theory for Smale spaces, why one might have been looking for such a theory and how the C*-algebras provide a key ingredient in the definition.

Mikael Rørdam (Copenhagen)
Elementary amenable groups have quasidiagonal C*-algebra

Rosenberg proved in 1987 that if the C*-algebra of a discrete group is quasidiagonal, then the group is amenable, and he conjectured that the converse also holds. Using techniques from the classification of C*-algebras and a description of elementary amenable groups due to Chou and Osin we confirm Rosenberg's conjecture for elementary amenable groups. We also show that these group C*-algebras are AF-embeddable. Our methods extend to show that also group C*-algebras of amenable LEF groups are AF-embeddable.

This is a joint work with N. Ozawa and Y. Sato.

Zhong-Jin Ruan (Illinois)
Exotic (Quantum) Group C*-algebras

Volker Runde (Alberta)
Dual Banach algebras - an overview

A dual Banach algebra is a Banach algebra that is also a dual Banach space such that multiplication is separately weak* continuous. Von Neumann algebras are dual Banach algebras, but so are the measure algebras of locally compact groups. We discuss amenability properties for dual Banach algebras as well as their surprisingly intricate representation theory.

Luis Santiago (Aberdeen)
TBA

Aaron Tikuisis (Aberdeen)
The dimension of approximately subhomogeneous C*-algebras
Coauthors : George Elliott, Luis Santiago, Wilhelm Winter

Viewed as a noncommutative topological space, it is quite sensible to try to define the dimension of a C*-algebra, and many fruitful concepts have arisen from doing so. I will focus on decomposition rank and ASH dimension, two dimension-like invariants which are important to the classification of C*-algebras. It is conjectured that for simple ASH algebras, the decomposition rank and the ASH dimension always agree and lie in the range $\{0,1,2,\infty\}$. This conjecture is closely related to the Toms-Winter conjecture and a revived Elliott conjecture. It is moreover corroborated by various results, some but not all of which use classification. I will discuss some new developments related to this conjecture.

Andrew Toms (Purdue)
Mean Dimension and crossed products

Wilhelm Winter (Münster)

Regularity of nuclear C*-algebras

 

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