WCOAS 97, Abstracts

Sixth West Coast Operator Algebra Seminar, October 1997

Abstracts of talks

The tree planar algebra and symmetries of Sunder subfactors
Bina Bhattacharyya
University of California, Berkeley

Abstract: I will define the "tree" planar algebras from the perspective of Jones' planar algebras and as endomorphisms of wreath product group representations. As a consequence of work of Popa and Jones, the tree planar algebras are standard invariants of a family of subfactors. I will also state their relationship to Sunder subfactors (subfactors arising from permutation biunitary matrices), which is the context in which they were discovered.

Noncommutative polyhedra and strong NF algebras
Bruce Blackadar
University of Nevada, Reno

Abstract: We consider inductive systems of finite-dimensional C*-algebras, where the connecting maps are complete order embeddings which are asymptotically multiplicative. This construction can be regarded as a noncommutative version of piecewise-linear topology, and we will exploit this point of view to study the structure of the inductive limit C*-algebras.

Random matrices and exact C*-algebras
Uffe Haagerup
Odense University

Abstract: The talk will be a report on a joint work with Steen Thorbjoernsen (Odense, Berkeley). Our main result is:

Let a₁,...,a_r be finitely many elements in an exact unital C*-algebra A, for which sum_i(a_i*.a_i) = 1 and ||sum_i(a_i.a_i*) ||= delta < 1. Assume that for every positive integer n, y₁(n),...,y_r(n) are r random n x n matrices, for which the entries (y_i(n))_k,l are n.r² independent comlex Gaussian random variables all with density n/pi times exp(-n.|z|²). Then the M_n(A)-valued random variable

S(n) = sum_i(a_i tensor y_i)
satisfies:
limsup_{n -> oo} sp(S(n)*.S(n)) =< (1 + sqrt(delta))²
and
liminf_{n -> oo} sp(S(n)*.S(n)) >= (1 - sqrt(delta))²

for almost all omega in the underlying probability space. In particular for almost all omega the operator S(n)*.S(n) becomes eventually invertible as n -> oo. The latter result is used to give a new proof of Blackadar's and Rordam's result, that states on K_o(A) are given by tracial states on A for any exact unital C*-algebra A. We will discuss the analytic and combinatorial aspects of the proof of the main result, and provide examples, which shows that both inequalities listed above are false for general non-exact C*-algebras.

The Hecke Algebra of Type B_n and Subfactors
Rosa Orellana
University of California, San Diego

Abstract: We determine all subfactors which can be obtained from the inclusion of Hecke Algebra of type A_n into Hecke algebras of type B_n+1 at roots of unity. They correspond to Young diagrams which contain a rectangle and can be related to certain subfactors constructed from Hecke algebras of type A. In particular, we compute their indices and higher relative commutants.

Joint similarity problems and the generation of operator algebras with bounded length
Gilles Pisier
Texas A & M and Universite Paris 6

Abstract: We will consider the following similarity problem: Let T₁, T₂ be two commuting operators which are each (separately) similar to a contraction, is the pair (T₁, T₂) similar to a pair of contractions? I.e. does there exist an invertible operator \xi such that both \xi^-1T₁\xi and \xi^-1T₂\xi are contractions? We show that the answer is negative.
There is an analogous phenomenon for pairs of unitarizable (uniformly bounded) representations on free groups. We will make the connection between such examples and the ``similarity degree" that we introduced recently in a broader context. In the present case of pairs, this leads naturally to a notion of ``length" for an operator algebra A with respect to two subalgebras A₁ \subset A and A₂ \subset A which (together) generate A. This notion is analogous to the length of a group with respect to a set of generators.

K-theory for factor maps between Cantor minimal systems
Ian Putnam
University of Victoria

Abstract: A Cantor minimal system is a compact, metrizable, totally disconnected space, X, with a homeomorphism, \phi, of X having no closed invariant sets except X and the empty set. Richard Herman, Christian Skau and I have provided a model for such systems using ordered Bratteli diagrams. We have also investigated the K-theory of the associated C*-algebra, denoted K*(X,\phi). In this talk, we consider a factor map \pi :(X,\phi ) -> (Y,\psi ) between two such systems. That is, \pi :X -> Y is continuous and \pi\circ\phi = \psi\circ\pi. Such a map induces a group homomorphism \pi*: K*(Y,\psi ) -> K*(X,\phi ). We discuss various properties of this map and its relation with \pi and we present several examples. (This is a joint work with Thierry Giordano and Christian Skau.)

Operator-valued semicircular systems
Dimitri Shlyakhtenko
University of California, Berkeley

Abstract: Operator-valued semicircular systems are the analogs of families of independent Gaussian families in operator-valued free probability theory. We investigate algebras generated by such systems (this class includes in particular the cores of free Araki-Woods factors). Among the applications of this language are some results on the cores of free Araki-Woods factors, as well as a result showing that for any II₁ factor M, the fundamental group F(M*L(F(infinity)) contains the fundamental group of M (generalizing the earlier results of Voiculescu and Radulescu on F(L(F(infinity)))).

Topics in free entropy
Dan Voiculescu
University of California, Berkeley

Abstract: The talk deals with recent work on free entropy: a new approach to free entropy based on a noncommutative Hilbert transform and some new results in the matricial microstates approach .

Applications of braided endomorphisms from conformal inclusions
Feng Xu
University of California, Los Angeles

Abstract: In this talk I will discuss the general theory of braided endomorphisms from conformal inclusions and some applications, including a negative solution to a well-known hypothesis (Kac-Wakimoto hypothesis) in the theory of Kac-Moody algebras which has existed for almost a decade.