Exact groups and C*-algebras
by
Simon Wassermann
University of Glasgow

EXACT GROUPS AND C*-ALGEBRAS

In the mid-1970s it became clear that a commonly used tensor product operation for C*-algebras can behave badly with respect to taking quotients. If A and B are C*-algebras faithfully represented on Hilbert spaces H and K, then the completion of the image of the algebraic tensor product A\odot B in L(H\otimesK) is the spatial tensor product A\otimesB. If I is a (two-sided, closed) ideal of B, so that the sequence

0 --> I --> B --> B/I --> 0

is exact, the sequence

0 --> A\otimesI --> A\otimesB --> A \otimes(B/I) --> 0

is, in general, not exact. If, for a particular A, this sequence is exact for arbitrary B and I, then A is said to be exact. Replacing A by a locally compact group, and the tensor products by reduced crossed products, one obtains, analogously, a notion of exactness for groups.

Over the last twenty years it has become apparent that the exact C*-algebras occupy a special position among all C*-algebras. There has been spectacular progress in understanding their structure, primarily as a result of the work of E. Kirchberg, one of whose most remarkable results is that a separable C*-algebra is exact if and only if it is a C*-subalgebra of a certain well-known simple C*-algebra, the Cuntz algebra O₂.

Recently, exact groups have attracted increasing attention, partly in connection with work on the Novikov conjecture. Kirchberg and the speaker have obtained various characterisations of exact groups; for example, a discrete group G is exact if and only if its regular C*-algebra C^*_r(G) is an exact C*-algebra. It has been shown that a wide class of locally compact groups, including most naturally occurring examples, are exact. One of the most perplexing questions, whether all discrete groups are exact, also appears to have been resolved at last. In recent months Ozawa has given a characterisation of group exactness which implies that certain groups whose construction has been announced by Gromov are not exact.

This talk will describe some of these developments.

Date received: June 14, 2000