Approximation by normal elements with finite spectra in C^∗-algebras of real rank zero

Huaxin Lin

Vol. 173 (1996), No. 2, 443–489

DOI: 10.2140/pjm.1996.173.443

Abstract

We study the problem when a normal element in a C^∗-algebra of real rank zero can be approximated by normal elements with finite spectra. We show that all purely infinite simple C^∗-algebras, irrational rotation algebras and some types of C^∗-algebras of inductive limit of the form C(X) ⊗ M_n of real rank zero have the property weak (FN), i.e., a normal element x can be approximated by normal elements with finite spectra if and only if Γ(x) = 0 (λ − x ∈ Inv ₀(A) for all λ∉sp(x)). For general C^∗-algebras with real rank zero, we show that a normal element x with dimsp(x) ≤ 1 can be approximated by normal elements with finite spectra if and only if Γ(x) = 0. One immediate application is that if A is a simple C^∗-algebra with real rank zero which is an inductive limit of C^∗-algebras of form C(X_n) ⊗ M_m(n), where each X_n is a compact subset of the plane, then A is an AF-algebra if and only if K₁(A) = 0.

Mathematical Subject Classification 2000

Primary: 46L05

Secondary: 46L80

Milestones

Received: 20 August 1993

Revised: 25 July 1994

Published: 1 April 1996