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

TL;DR

This paper studies modules over C*-algebras equipped with a norm-like map, revealing their structure and showing that non-commutative cases are Hilbert modules while commutative cases relate to bundles of Banach spaces.

Contribution

It characterizes modules with norm-like maps over C*-algebras, distinguishing between commutative and non-commutative cases and describing their structure.

Findings

Modules over non-commutative C*-algebras are Hilbert modules.

Modules over commutative C*-algebras correspond to sections of Banach bundles.

General modules embed into sums of the two types.

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 = C_0(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.