Unitary equivalence modulo the compact operators and extensions of C*-algebras

Introduction Notation, definitions and useful facts Amenable traces and stronger approximation properties Examples and special cases Finite representations Applications and connections with other areas Bibliography.

Invariant Means and Finite Representation Theory of $c*$-Algebras

Every discrete group G generates a C*-algebra 9Q1(G) of operators on the Hilbert space L2(G) of square summable complex valued functions on G. 9Q1(G), in which elements are written as formal (generally infinite) complex linear combinations ExE -GaXx, is the closure in operator norm of the group algebra L(G) of finite linear combinations, which acts on L2(G) by left multiplication (the left regular representation of G). Our main result is a formula for the explicit calculation of the norm of certain operators in L(G). We show that, if xl, . . ., x, satisfy a certain "freedom condition" (in particular if they freely generate a free subgroup of G), then for arbitrary complex scalars al,,... ,an5 a~ ~ ~ ~ ~~~Z Z21 i1 _Z JZ>A7

Computing norms in group c*-algebras.*

We prove that it is relatively consistent with the usual ax- ioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips{Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown{Douglas{Fillmore. We also give a simpler and self-contained proof of the Phillips{Weaver result.

http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n2-p01-p.pdf

All automorphisms of the calkin algebra are inner

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC *-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA 2(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA 2(C N),N≧1.

B.E. Johnson

Papers

Operators commuting with a von Neumann algebra modulo the set of compact operators