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

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.