A note on power bounded elements of von neumann algebras

Abstract

Let A be a von Neumann algebra with predual A(*) and the unit element 1. An element a of A is said to be power bounded if sup(n >= 0) vertical bar vertical bar a(n)vertical bar vertical bar < infinity. In this note we show that, for any power bounded element a of A, theta = sigma(A, A(*)) - lim(n ->infinity)(1+a/2)(n) exists, theta is an idempotent and theta a = a theta = theta.