Talks and Abstracts

For a given countable group Γ we consider the following three properties:
1. Γ has an infinite subgroup with relative property (T).
2. The group von Neumann algebra L(Γ) has a diffuse von Neumann subalgebra with relative property (T).
3. Γ does not have Haagerup's property.
It is clear that (1) ⇒ (2) ⇒ (3). We prove that both of the converses are false. This is joint work with Adrian Ioana.

A brief survey of what is known concerning simple C*-algebra inductive limits of matrix algebras over compact metric spaces of unbounded dimension is given. The first such examples, which could not be obtained using bounded dimension, were given by Jesper Villadsen in his Ph.D. thesis. The question of extending the known classification result in the case of bounded dimension (due to Gong, Li, and the speaker) to the case of unbounded dimension is considered. It has been shown by Andrew Toms that the Cuntz semigroup is needed (in addition to the more usual invariants) to distinguish algebras in this class. It is not clear what other invariants will be needed.

An E₀-semigroup is a one-parameter semigroup of endomorphisms of B(H). We consider families of E₀-semigroups parametrized by a compact Hausdorff space X, with the appropriate continuity requirement. We provide a classification of such perturbed families in terms of an invariant given by vector bundles over the space X, in the case in which all E₀-semigroups in the family are of type I_n for a given n.
Joint work with Daniel Markiewicz.

We say that a C*-algebra A is Fell (or type I₀) if it is generated by abelian elements. In this case A is almost a continuous trace algebra but  need not be Hausdorff. Such algebras arise naturally in the study of certain dynamical systems. We prove:
• An abelian C*-subalgebra B of a type I₀ algebra A is a diagonal iff it satisfies the extension property. (i.e. pure states of B extend uniquely).
• Up to Rieffel-Morita equivalence (RME) each such A contains a diagonal.
• The twists arising from RME algebras of type I₀ containing diagonals are equivalent in a natural sense.
This opens the door for a classification of such algebras up to RME.
Joint work with Astrid an Huef and Aidan Sims.

Suppose that E_i is a C*-correspondence over the C*-algebra A_i, i = 1,2. A (strong) Morita equivalence between (A₁,E₁) and (A₂,E₂) is an invertible C*-correspondence X from A₁ to A₂ such that E₁ ⊗ _A1 X ≅ X ⊗ _A2 E₂. In Proc. London Math. Soc. 81 (2000), 113-168, we showed that a Morita equivalence between (A1,E1) and (A2,E2) induces a strong Morita equivalence between the corresponding tensor algebras T₊(E₁) and T₊(E₂) in the sense of Blecher, Muhly and Paulsen in the Memoirs of the AMS 143 (2000), no. 681. In this talk we will make precise the sense in which a strong Morita equivalence between (A1,E1) and (A2,E2) induces an isometry between the space of completely contractive representations of T₊(E₁) and the completely contractive representations of T₊(E₂) and discuss other features of the representation theory of tensor algebras that are preserved under this notion of Morita equivalence.
Joint work with Baruch Solel.

The extended Haagerup subfactor was the last unknown item on Haagerup's 1993 list of possible small-index subfactors. We construct this subfactor by constructing its associated planar algebra. This nishes the classication of subfactors with index up to 3 + √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.
Joint work with Stephen Bigelow, Scott Morrison and Noah Snyder.

I will present a general setting to prove U_fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting I will provide new examples of this phenomenon, extending results of S. Popa. I will also use a result of K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be U_fin-cocycle superrigid. This is joint work with Thomas Sinclair.

Let X be a finite dimensional compact metric space, and let h: Z^d × X → X be a free minimal action. We describe initial work towards understanding the structure of the transformation group C*-algebra C*(Z^d,X,h).

We consider II₁ factors M which can be realized as inductive limits of subfactors, N_n ↑ M, having spectral gap and satisfying the bi-commutant condition (N_n'∩M)' ∩ M = N_n. Examples are the enveloping algebras associated to non-Gamma subfactors of finite depth, as well as certain crossed product algebras. We use deformation/rigidity techniques to obtain classification results for such factors and to calculate Connes' invariant χ(M).

For (discrete) directed graphs (and subsequently for higher-rank graphs), Raeburn et al developed a satisfying theory of coverings and fundamental groups. The coverings were closely related to skew products, and the associated C*-algebras turned out to be crossed products by coactions. In joint work with Valentin Deaconu and Steve Kaliszewski, we are (in the process of) developing a version of this theory for the topological graphs of Katsura. The noncommutative duality seems to carry over, but since the groups are no longer discrete the coverings become something else.

A field of Hilbert spaces may be expressed as a supremum of locally trivial vector bundles defined on open subsets of the base space. This point of view may be exploited to transplant results from the theory of vector bundles to the setting of fields of Hilbert spaces. For example, one can always embed a field of Hilbert spaces inside another one with sufficiently larger dimension (depending on the covering dimension of the base space). One can use clutching functions to construct new fields of Hilbert spaces from old ones. If the base space has dimension at most 3, all the isomorphism classes of fields of Hilbert spaces may be described in terms of cohomological data. I will talk about these and other results obtained recently in collaboration with Aaron Tikuisis.

We will show how the homomorphisms from the C*-algebra of continuous function on a tree to a C*-algebra of stable rank one can be classified by means of the Cuntz functor. In the special case when the tree consists of a single edge we describe a class of codomain C*-algebras - not necessarily of stable rank one - for which this classification holds. We will also discuss how in certain cases the classification fails. These results are obtained in a joint work with Alin Ciuperca and George Elliott and in a joint work with Leonel Robert.

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II₁ factor 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.
Joint work with Cyril Houdayer.

We study another q-deformation of Brauer's centralizer algebra. It contains the Hecke algebras of type A as a subalgebra, where the embedding is determined by certain commuting square conditions. This is motivated by the problem of finding a rigorous construction of subfactors in connection with twisted loop groups. We also give formulas for indices and relative commutants of such subfactors.