On operator bands

A multiplicative semigroup of idempotent operators is called an operator ba nd. We prove that for each K > 1 there exists an irreducible operator band on the Hilbert space l(2) which is norm-bounded by K. This implies that the re exists an irreducible operator band on a Banach space such that each mem ber has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator ban d on l(2) that is weakly r-transitive and is not weakly (r + 1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A, B epsilon S, where p is a given polynomial in two non- commuting variables. It turns out that the polynomial p(A, B) = (AB - BA)(2 ) has a special role in these considerations.