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 < ∞ , we also prove an analogue of the result that trace ( A B ) = trace ( B A ) for a trace class operator A and a bounded operator B on a Hilbert space.