Functional decompositions on vector-valued function spaces via operators

Let X be a Banach space with a generalized basis. The Banach algebra B(X) of bounded linear operators on X is used to construct Banach spaces, M and K, of weak* continuous functions from the state space of a C*-algebra to B(X). If the basis satisfies certain properties, we prove that the dual space of M has a decomposition analogous to that of the dual space of B(X). In terms of the notion of M-ideal introduced by Alfsen and Effros, the subspace K is an M-ideal in the Banach space M. For the cases of c(0) and l(p), 1 < p < infinity, we also prove an analogue of the result that trace(AB) = trace(BA) for a trace class operator A and a bounded operator B on a Hilbert space. (C) 2012 Elsevier Inc. All rights reserved.