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 c0 and ℓp, 1