The topic of the present paper is concrete Banach and C*-algebras which are generated by a finite number of idempotents. Our first result is that, for each finitely generated Banach algebra A, there is a number n 0 so that the algebra A n_n of all n_n matrices with entries in A is generated by three idempotents whenever n n 0 , and that A n_n is generated by two idempotents if and only if n=2 and if A is singly generated. As an application we find that the algebra C n_n (K) of all continuous C n_n -matrix-valued functions on a compact K/C with connected complement but without interior points, is generated by 2 or 3 idempotents in case n=2 or n>2, respectively. This result is used to construct examples of C*-algebras which are generated by 2 idempotents but not 2 projections. For these algebras, the standard 2_2 matrix symbol fails to be symmetric. We finally show that each C*-algebra satisfying a polynomial identity (in particular, each C*-algebra generated by two idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one. 1996 Academic Press, Inc.
1. Finitely Generated Banach Algebras, and Idempotents
An element a of an algebra over the complex field C is said to be idempotent if a 2 =a. A projection is an element of an involutive complex algebra satisfying a 2 =a and a*=a. (This notation is justified because each C*-algebra is *-isomorphic to a symmetric subalgebra of the algebra of all article no.