Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra A there e xists 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 statem ent is true for direct sums (A) over tilde = M-n1(A) + M-n2(A) + ... + M-np (A) and (B) over tilde = M-n1(B) + M-n2(B) + ... + M-np(B) (n(j) > 1), wher e B is a finitely generated free algebra, i.e. polynomials in several non-c ommuting 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 (A) over tilde 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 M-2(B) can not b e generated by two idempotents. But (B) over tilde can be generated by thre e idempotents for eadl singly generated free algebra B.