Compact homomorphisms of 𝐶*-algebras

Abstract

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. . Primary 46K05, 46L05, 47B05; Secondary 43A65, 43A75.Key words and phrases. Compact homomorphism, finite rank operator, group of unitary elements, semisimple Banach algebra, simple pole, spectral idempotent.