Compact endomorphisms and closed ideals in Banach algebras

Abstract

Abstract.Every infinite-dimensional Banach algebra with a nonzero compact endomorphism has a proper closed nonzero two-sided ideal. When the algebra is commutative, the ideal is also an ideal in the multiplier algebra.Suppose that A is a Banach algebra with a compact nonzero endomorphism. We show that A has a proper closed nonzero two-sided ideal, and we prove a slightly stronger result when A is commutative. There are many Banach algebras which have compact endomorphisms [5]; this can even happen for commutative radical integral domains [3, §5]. On the other hand, some primitive Banach algebras, such as the algebra of compact operators on Hilbert space, have no nonzero two-sided ideals. Our results show that algebras with no nonzero closed ideals can have no compact nonzero endomorphisms.Our proofs involve applying extensions, which we developed in [4], of Lomonosov's invariant subspace theorems. Apparently the best that one can do with the original versions of Lomonosov's theorem [6,7] is to show that if A has a compact left or right multiplication operator, then it has both left and right nonzero closed ideals. Using very different methods, Esterle [1] has recently made significant contributions to the study of the problem of whether every radical Banach algebra has a nonzero closed ideal. Theorem 1. // the infinite-dimensional Banach algebra A has a compact nonzero endomorphism <j>, then A has a proper closed nonzero ideal.Proof. Let A9 be the Banach algebra formed by adjoining an identity to A if A has no identity, and let A* = A if A has an identity. For each a in A%, let La be the left multiplication operator Lax = ax, and leti? be the algebra of all left multiplication operators La on A for a in A*. Similarly, let ¿% be the algebra of right multiplication operators. To show that A has a proper closed ideal, we show that ££ and 0t have a common proper invariant subspace. We do this by showing that the algebrase and i% satisfy the hypotheses of [4, Theorem (3.2), p. 849].The map a -» La is a continuous linear transformation from A* to the algebra of bounded operators on A. Hence =5? is an operator range algebra in the sense of [4, p. 845] (cf. [2, §3]). Since <f> is a homomorphism, we have <t>La = L^a)^ for every a in

Compact homomorphisms of 𝐶*-algebras

“…The existence of a nonzero compact endomorphism on a Banach algebra implies the existence of a nonzero proper closed two-sided ideal in that algebra, as S. Grabiner has shown in [4]. Throughout this paper "homomorphism" will mean an "algebra homomorphism."…”