Central decompositions for compact convex sets
A Banach algebra A is called strict if the product morphism is continuous with respect to the weak norm in A ⊗ A. The following result is proved: A C*-algebra is strict if and only if all its irreducible representations are finite-dimensional and their dimensions are bounded.
Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting A = TA ⊕ kerT are listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA ⊕ kerT = A if and only if there is a commuting operator S for which T = TST and S = STS, that this is the case if and only if such S may be taken to be a multiplier, and that these conditions are also equivalent to the existence of a factorization T = PB,...