Compact and weakly compact derivations of C^∗-algebras

Charles A. Akemann and Steve Wright

Vol. 85 (1979), No. 2, 253–259

DOI: 10.2140/pjm.1979.85.253

Abstract

In a forthcoming paper, the second-named author asks if every compact derivation of a C^∗-algebra 𝒜 into a Banach 𝒜-module X is the uniform limit of flnite-rank derivations. We answer this question affirmatively in the present paper when X = 𝒜 by characterizing the structure of compact derivations of C^∗-algebras. In addition, the structure of weakly compact derivations of C^∗-algebras is determined, and as immediate corollaries of these results, necessary and sufficient conditions are given for a C^∗-algebra to admit a nonzero compact or weakly compact derivation.

Mathematical Subject Classification 2000

Primary: 47B47

Secondary: 46L05

Milestones

Received: 17 November 1978

Published: 1 December 1979