Topological Rees Matrix Semigroups

Abstract

An important problem in the theory of topological semigroups is to formulate a suitable definition of continuity for the choice of generalized inverses. In this paper, we will show that under certain natural conditions, a topology can be defined on a Rees matrix semigroup, which turns it into a topological semigroup, and in which a canonical continuous choice of inverses is possible. As an example, we show that this construction applied to the semigroup of operators of rank less than or equal to 1 on a Hilbert space gives a topology which is stronger than the norm topology, under which this semigroup is a topological semigroup and the assignment of every operator to its Moore-Penrose inverse is continuous.