OnC*-Algebras Generated by Idempotents

The topic of the present paper is concrete Banach andC*-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 numbern0so that the algebra An×nof alln×nmatrices with entries in A is generated by three idempotents whenevern⩾n0, and that An×nis generated by two idempotents if and only ifn=2 and if A is singly generated. As an application we find that the algebraCn×n(K) of all continuous Cn×n-matrix-valued functions on a compactK⊂C with connected complement but without interior points, is generated by 2 or 3 idempotents in casen=2 orn>2, respectively. This result is used to construct examples ofC*-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 eachC*-algebra satisfying a polynomial identity (in particular, eachC*-algebra generated by two idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one.