A history of topological and analytical semigroups: a personal view

Abstract

and what nowadays, in more general form, is called domains, whileJ. D. LAWSON drew semigroup theoreticians' attention to a very natural class of compact semilattices having enough homomorphisms into the unit interval semilattice. The class of continuous lattices agrees with the class of Lawson semilattices. It generates a network of applications in theoretical computer science under the name "domain theory". - A hundred years after SOPHUS LIE's differentiable groups and semigroups, attention returned back to semigroups and Lie theory. Lie semigroup theory, initiated by E. B. VINBERG, G. I. OLSHANSKY, J. D. LAWSON and the author among others, infused a strong geometric and analytical flavor into topological semigroup theory and generated a new lines of application of semigroup theory such as in geometric control theory, and in the area of unitary representation theory of Lie groups, particulary in the area of holomorphic extensions of unitary representations. A respectable number of mongraphs and collections have been and are being written in this field.