# Applications of amenable semigroups in operator theory

## TL;DR

This paper explores how amenable semigroups can be represented as bounded linear operators on Banach spaces, providing conditions for projections, isometries, and operator extensions relevant to operator theory.

## Contribution

It introduces new sufficient conditions for projections and invertible operators commuting with semigroup representations, advancing understanding in operator theory on Banach and Hilbert spaces.

## Key findings

- Conditions for projections commuting with semigroup actions
- Criteria for invertible operators making semigroup elements isometries
- Results on extending intertwining operators and renorming

## Abstract

The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on operators on hereditarily indecomposable Banach spaces are offered.