Abelian Strict Approximation in Multiplier C^*-algebras
by
László Zsidó
University of Rome ``Tor Vergata"
Coauthors: Claudio D'Antoni (University of Rome ``Tor Vergata")

Let A be a C^*-algebra and M(A) its multiplier algebra. Our aim is to investigate the richness of the union of the strict closures in M(A) of all commutative C^*-subalgebras of A, called in the sequel the abelian strict closure of A. We notice that the classical Weyl-von Neumann-Berg-Sikonia theorem claims that, for A the compact operators on a separable Hilbert space, every normal element of M(A) belongs modulo A to the abelian strict closure of A.

Among others we prove:

1) Assume that A is sigma-unital and B is a separable C^*-subalgebra of M(A). Then there are separable C^*-subalgebras B₁, B₂ of M(A), contained modulo A in the hereditary C^*-subalgebra of M(A) generated by B, such that every normal element of B₁ and B₂ belongs to the abelian strict closure of A and B is contained in A + B₁ + B₂.

2) Assume that M(A) is a type II_\infty von Neumann algebra and A is the norm closed linear span of all finite projections of M(A). Then the abelian strict closure of A is contained in A.

Date received: April 27, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-31.