Continuous Bounded Cohomology of Topological Semigroups

TL;DR

This paper investigates the properties of continuous bounded cohomology groups in topological semigroups, establishing new results about their structure, triviality in certain cases, and applications to topological lattices.

Contribution

It provides new insights into the structure of continuous bounded cohomology groups of topological semigroups, including Banach space properties and triviality results for amenable cases.

Findings

Second cohomology group of compact metrizable semigroup is a Banach space

Cohomology groups of rank > 1 are trivial for compact amenable semigroups

Examples and applications to topological lattices

Abstract

In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.