MODULES WITH NORMS WHICH TAKE VALUES IN A C-ASTERISK-ALGEBRA

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