Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra A there exists a number N such that for each n > N the matrix algebras M_n(A) can be generated by three idempotents. In this paper we show that the same statement is true for direct sums à = M_n1 (A) ⊕ M_n2 (A) ⊕ ... ⊕ M_np (A) and B̃ = M_n1 (B) ⊕ M_n2 (B) ⊕ ... ⊕ M_np (B) (n_j > 1) , where B is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras M_n(A) because the number N we obtain here improves known estimates (see for example [R]). We show that the algebra à can be generated by two idempotents if and only if n_j = 2 for each j and A is singly generated. Also we give an example of a free singly generated algebra B for which M₂(B) can not be generated by two idempotents. But B̃ can be generated by three idempotents for each singly generated free algebra B.