Let A be a semi-simple
annihilator Banach algebra, and let ν be a homomorphism of A into a Banach
algebra. In this paper we describe various continuity properties of ν. Let (∗) be
the condition that I ⊕R(I) = A for all closed two-sided ideals I, where
R(I) = {x∣Ix = (0)}. If (∗) holds, then we show that there exists a constant K and a
finite set of primitive ideals such that ∥ν(x)∥≦ K∥x∥⋅∥y∥ whenever yx = x and x is
in the intersection of this finite set. If (∗) does not hold, then essentially the same
conclusion is true, but with the given norm replaced by one which is defined on a
dense subset of A. If A is a dual B∗-algebra, then ν is continuous on the
socle.
We also consider the existence of unconditional decompositions in A. We show
that (∗) holds if and only if the minimal-closed two-sided ideals of A form an
unconditional decomposition for A.
