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Tensor products of compact convex sets and Banach algebras

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.