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 script capital A sign, there is a number n₀ so that the algebra script capital A sign^n×n of all n × n matrices with entries in script capital A sign is generated by three idempotents whenever n ≥n ₀, and that script capital A sign^n×n is generated by two idempotents if and only if n = 2 and if script capital A sign 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.