Modules with norms which take values in a C*-algebra

N.C. Phillips and N. Weaver

Vol. 185 (1998), No. 1, 163–181

DOI: 10.2140/pjm.1998.185.163

Abstract

We consider modules E over a C*-algebra A which are equipped with a map into A₊ that has the formal properties of a norm. We completely determine the structure of these modules. In particular, we show that if A has no nonzero commutative ideals then every such E must be a Hilbert module. The commutative case is much less rigid: If A = C₀(X) is commutative then E is merely isomorphic to the module of continuous sections of some bundle of Banach spaces over X. In general E will embed in a direct sum of modules of the preceding two types.

Milestones

Received: 18 December 1996

Revised: 1 April 1997

Published: 1 September 1998