FINITE-DIMENSIONAL IDEALS IN BANACH ALGEBRAS BY

Let A be a semi-prime Banach algebra. By an ideal in A we shall always mean a two-sided ideal unless otherwise specified. Smyth [9] has shown that, for x in A, xA is finite-dimensional if and only if Ax is finite-dimensional. Let F be the set of all x in A for which xA is finite-dimensional. We extend Smyth’s theorem as follows. Let K be any ideal in A. Then, for x in A, xK is finite-dimensional if and only if Kx is finite-dimensional. Note that a distinction between this result and the Smyth case where K = A is that x need not lie in K. Then we describe and study F and its role in Banach algebra theory. Let Γ be the set of non-zero central idempotents p in the socle of A for which pA is a simple algebra. All these are in F and F is the direct sum of the ideals pA for p in Γ. In the theory of commutative Banach algebras much attention is devoted to seeing when an ideal must be contained in a modular maximal ideal. We consider the non-commutative case where A has a dense socle and B is the completion of A in some normed algebra norm on A. An ideal W of B is contained in a modular maximal ideal of B if and only ifW does not contain F. Easy examples show this can fail if A does not have a dense socle. First we treat some preliminaries. Throughout A is a semi-prime Banach algebra over the complex field with socle S and center Z. For an ideal