WCOAS09 - Abstracts

Talk Abstracts
Marius Dadarlat
E-theory for C*-algebras over a space. Abstract: We introduce E-theory for separable C*-algebras over possibly non-Hausdorff topological spaces and establish its basic properties, such as six-term exact sequences in each variable and a universality property. We also plan to discuss certain applications. (This is joint work with Ralf Meyer).
Calder Daenzer
A groupoid approach to orbifold Picard torsors. Abstract: I will describe a fairly simple construction with groupoids of ``orbifold Picard torsors.'' These are a common generalization of a principal bundle, a gerbe, and a family of orbifold groups. I will give several examples, and show how such objects may be useful in theoretical physics.
Franz Luef
Projective modules over noncommutative tori and Gabor analysis. Abstract: In this talk I present an interpretation of projective modules over noncommutative tori in terms of Gabor analysis, which is a modern branch of time-frequency analysis. Furthermore I will discuss spectrally invariant subalgebras of the noncommutative torus that are different from the smooth noncommutative torus and the construction of projective modules over these spectrally invariant subalgebras. These results rely on a class of function spaces, so called modulation spaces. In particular, I will explain the relation between the modulation spaces and spectrally invariant subalgebras of the noncommutative torus. Finally I want to draw some consequences from this fact about the structure of Gabor frames, and about the construction of projections in noncommutative tori.
Emily Peters
The extended Haagerup planar algebra. Abstract: The extended Haagerup subfactor was the last unknown item on Haagerup's 1993 list of possible small-index subfactors. In recent work with Stephen Bigelow, Scott Morrison and Noah Snyder we construct this subfactor by constructing its associated planar algebra. This finishes the classification of subfactors with index up to $3+\sqrt{2}$. Our construction works by identifying a planar subalgebra of the graph planar algebra of the desired principal graph. The challenge is to demonstrate that this planar subalgebra is small enough to be a subfactor planar algebra, which we accomplish by viewing some of the relations on the subalgebra as substitutes for a braiding relation.
Arlan Ramsay
Fourier-Stieltjes algebras of locally compact groupoids (joint work with A. Paterson).
Jean Renault
Cartan subalgebras in the non Hausdorff case (joint work with A. Buss and R. Exel). Abstract: A standard construction associates to a pseudogroup of partial homeomorphisms of a locally compact Hausdorff space a C$^*$- algebra. When the groupoid of germs of the pseudogroup is Hausdorff, the C$^*$-algebra which arises in this fashion can be characterized by the existence of a nice maximal abelian self-adjoint subalgebra, called a Cartan subalgebra. More precisely, this construction establishes an equivalence of categories between twisted \'etale locally compact Hausdorff effective groupoids and Cartan pairs. This is a C$^*$-algebraic analogue of a well-known theorem of J.~Feldman and C.~Moore on Cartan subalgebras in von Neumann algebras. I will report on the non-Hausdorff case.
Sam Schmidt
Endomorphisms, the Toeplitz Algebra and Composition Operators. Abstract: Local homeomorphims of the unit circle induce endomorphisms of the continuous functions on the unit circle, $C(\mathbb{T})$. By studying the composition operator associated with certain local homeomophisms one can determine when an endomorphism will extend to the Toeplitz algebra, $\mathcal{T}$. That is, given a local homeomorphism $\sigma: \T \rightarrow \T$ when does there exist a $*$-endomorphism $\tilde{\sigma}$ of $\mathcal{T}$ so that $\tilde{\sigma}(\tau(f) + k) = \tau( f \circ \sigma) + k^{\prime}$ where $k^{\prime}$ is some other compact operator? We will explore this and related questions.
Dimitri Shlyakhtenko  Strongly solid II1 factors with an exotic MASA (joint work with C. Houdayer). Abstract: Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1factor M containing an ”exotic” maximal abelian subalgebra A: as an A,A-bimodule, L2(M) is neither coarse nor discrete. Thus we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu’s free entropy results that M is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.
Aidan Sims
Algebras defined by co-universal properties. Abstract: In areas such as graph $C^*$-algebras, Cuntz-Pimsner algebras, crossed product $C^*$-algebras, and $C^*$-algebras of inverse semigroups, the initial objective is typically to identify a set of generators and relations which determine a ``suitable" $C^*$-algebraic realisation of the base object. The word ``suitable" is used loosely, but is usually taken to mean that the universal $C^*$-algebra satisfies a version of the gauge-invariant uniqueness theorem: every representation which is nonzero on generators and preserves a canonical gauge-action (typically of $\mathbb{T}$) is injective. In this talk we will discuss an approach to formalising the use of the word ``suitable" using the concept of a co-universal property inspired by Katsura's work on $C^*$-algebras associated to Hilbert bimodules. We will discuss some examples which indicate the advantages of this approach. This includes joint work with Carlsen, Larsen and Vittadello and work in progress with Exel and with Brownlowe and Vittadello.
Andrew Toms
Ranks of operators in simple C*-algebras. Abstract: Each (quasi-)trace on a C*-algebra gives rise to a dimension function---a way of measuring the rank of an operator with respect to the given trace. The collection of all such then associates to each operator a function on the trace space. We consider a basic question: which functions on the trace space arise in this way? We show that the answer is "all of them, subject to the obvious constraints" for several classes of simple C*-algebras. These results are then applied to give Z-stability and classification-by-K-theory results for the algebras considered. Much of the talk is based on joint work with M. Dadarlat.
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