On C*-algebras generated by idempotents

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(o) so that the algebra A(nxn) of all n x n matrices with entries in A is generated by three idempotents whenever n greater than or equal to n(o), and that A(nxn) 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-nxn(K) of all continuous C-nxn-matrix-valued functions on a compact K subset of 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 x 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. (C) 1996 academic Press, Inc.