Compact Homomorphisms of (t-algebras

Suppose A is a C*-algebra and B is a Banach algebra such that it can be continuously imbedded in B(H), the Banach algebra of bounded linear operators on some Hubert space H. It is shown that if 6 is a compact algebra homomorphism from A into B, then 6 is a finite rank operator, and the range of 0 is spanned by a finite number of idempotents. If, moreover, B is commutative, then 9 has the form 8{x) = xi(x)Ei + ■ • • + Xk{x)Ek, where Ei,... ,Ek are fixed mutually orthogonal idempotents in B and x 11 ■ • • > Xfc aie fixed multiplicative linear functionals on A. Introduction. Suppose A is a commutative, semi-simple, unital Banach algebra. In [8] H. Kamowitz proved that if 6 is a compact endomorphism on A, A' is the set of all multiplicative linear functionals on A, and 9* is the adjoint of 9, then f] 6*" (A1) is finite. One consequence of this result is a characterization of compact endomorphisms of C(X) [8, Corollary 2.2]. In this paper we characterize compact homomorphisms in a more general setting where they are defined from a C*-algebra into a Banach algebra that has a continuous imbedding in B(H). 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." We call a set of idempotents {e¿ : i G 1} mutually orthogonal if e¿ej = 0, whenever i í jTo prove our main result we will need the following extension lemma. LEMMA. Let A be a C*-algebra without identity, and let 6 be a compact homomorphism from A into a Banach algebra B. Then, there exists a compact homomorphism 9 from the C* -unitization C ® A of A into B which extends 9. PROOF. Let (ea) be a bounded approximate identity of A with supQ ||eQ|| < 1, [2, Theorem 12.4]. By the compactness of 9, there exists a subnet (0(ea,)) of (9(ea)) and an element e G B, such that 9(ea¡) —► e, in norm. Then for every x G A, we have e9(x) = lim9(eai)9(x) = lim0(eQ,x) = 9(x), 6(x)e = lim 6(x)0(eai ) = lim 9(xea, ) = 0(x). Received by the editors November 6, 1986 and, in revised form, March 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46K05, 46L05, 47B05; Secondary 43A65, 43A75.