On the subalgebra of a Fourier-Steiltjes algebra generated by pure positive definite functions

TL;DR

This paper investigates the algebraic structure of the subspace generated by pure positive definite functions in Fourier-Stieltjes algebras for various groups, revealing that it is not always an algebra and exploring related properties.

Contribution

It demonstrates that the subspace generated by pure positive definite functions is not necessarily an algebra and analyzes its structure and amenability properties.

Findings

In some groups, the subspace is an algebra; in others, it is not.

The structure of these subspaces varies significantly across different groups.

Properties related to amenability are examined within these subalgebras.

Abstract

For a locally compact group $G$, the first-named author considered the closed subspace $a_0(G)$ which is generated by the pure positive definite functions. In many cases $a_0(G)$ is itself an algebra. We illustrate using Heisenburg groups and the $2\times 2$ real special linear group, that this is not the case in general. We examine the structures of the algebras thereby created and examine properties related to amenability.