Vol. 25, No. 2, 1968

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Homomorphisms of annihilator Banach algebras

Gregory Frank Bachelis

Vol. 25 (1968), No. 2, 229–247

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) = {xIx = (0)}. If () holds, then we show that there exists a constant K and a finite set of primitive ideals such that ν(x)Kx∥⋅∥ywhenever 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.

Mathematical Subject Classification
Primary: 46.50
Received: 10 November 1966
Published: 1 May 1968
Gregory Frank Bachelis