New deformations of Convolution algebras and Fourier algebras on locally compact groups

TL;DR

This paper introduces novel deformations of convolution and Fourier algebras on locally compact groups, revealing structural insights about the groups through Banach algebra properties and operator algebra representations.

Contribution

It presents new deformation methods for convolution and Fourier algebras, linking algebraic properties to the underlying group's structure and growth characteristics.

Findings

Deformed convolution algebras on compact Lie groups relate to the group's real dimension.

Deformed Fourier algebras on finitely generated groups connect to the group's growth rate.

Deformations enable analysis of groups via Banach and operator algebra properties.

Abstract

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some informations of the underlying groups by examinining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similary, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.