On convoluters on $L^p$-spaces

TL;DR

This paper investigates convolution operators on $L^p$-spaces over locally compact groups, establishing key algebraic characterizations of convoluters and pseudo-measures under certain group properties.

Contribution

It proves that for groups with the approximation property, the algebra of convoluters equals the algebra of pseudo-measures, and describes the bicommutant of pseudo-measures.

Findings

Convoluters algebra equals pseudo-measures algebra for groups with the approximation property.

The bicommutant of pseudo-measures algebra is the convoluters algebra.

Provides algebraic characterizations of convolution operators on $L^p$-spaces.

Abstract

We prove two theorems about convolution operators on $L^p(G)$ for a locally compact group $G$. First, if $G$ has the approximation property, then the algebra of convoluters is the algebra of pseudo-measures. Second, the bicommutant of the algebra of pseudo-measures is the algebra of convoluters.