Modules with norms which take values in a \(C^*\)-algebra (Q1366650)

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.