A simple proof of Brown's diagonalizability theorem

Abstract

We present here a simple proof of Brown's diagonalizability theorem for certain elements of the algebra of a left regular band, including probability measures.

This note was written in October 2009. At the moment I have no intention to publish it since I intend to later publish a generalization to a larger class of semigroups, for which the proof is more complicated. I decided now to place it on ArXiv because it seems people are still interested in Brown's theorem and I believe this proof is the simplest one although it does not yield the eigenvectors