Minimal number of idempotent generators for certain algebras (Q5926412)

Let \(A\) be a Banach algebra with unit generated by \(k\) elements and \(M_n (A)\) be the matrix algebra. Let also \(\nu (M_n (A))\) be a minimal number of idempotents generating this algebra. It is known that for \(n\geq k+2\) last algebra can be generated by three idempotents. More precisely \[ \nu (M_n (A)) \leq 3 \;(n\geq 2) \;\text{for all} n\geq 1+2\sqrt{k-1} . \tag{1} \] The authors show that (1) holds for \(n\geq 2\sqrt{k-1}\). The same assertion is also considered for direct sum \(M_{n_1}(A)\oplus M_{n_2}(A)\oplus \dots\oplus M_{n_p}(A)\) and \(M_{n_1}(B)\oplus M_{n_2}(B)\oplus \dots\oplus M_{n_p}(B)\), where \(B\) is a finitely generated free algebra and \(n_j \geq 2\).