Journal of Operator Theory

Volume 49, Issue 1, Winter 2003 pp. 99-113.

Abelian strict approximation in multiplier $C^*$-algebras and related questions

Authors: Claudio D'Antoni (1) and Laszlo Zsido (2)
Author institution: (1) Dipartimento di Matematica, Universita di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italia

Summary: We prove a general result on the strict approximability of normal elements of the multiplier algebra $M(A)$ of a $\sigma$-unital $C^*$-algebra $A$ from commutative $C^*$-algebras of $A $. As an application, we reprove a result of L.G. Brown concerning the non-existence of non-zero separable hereditary $C^*$-algebras of the corona algebras of $\sigma$-unital $C^*$-algebras. Subsequently we characterize the situation in which an $W^*$-algebras (whose class contains all corona algebras of $\sigma$-unital $C^*$-algebras) allows non-zero separable hereditary $C^*$-subalgebras.

Keywords: $C^*$-algebra, multiplier algebra, strict topology, hereditary $C^*$-subalgebra, $SAW^*$-algebra, $AW^*$-algebra

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