## Abstract

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 Ã = 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 Ã 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 be generated by two idempotents. But B̃ can be generated by three idempotents for each singly generated free algebra B.

Original language | English |
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Pages (from-to) | 20-31 |

Number of pages | 12 |

Journal | Integral Equations and Operator Theory |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |