Convolutions on the Haagerup tensor products of Fourier algebras

TL;DR

This paper investigates convolution maps on the Haagerup tensor product of Fourier algebras of compact groups, revealing new insights into their structure, spectral synthesis, and relationships with tensor product factorizations.

Contribution

It provides a detailed analysis of convolution maps on Haagerup tensor products of Fourier algebras, highlighting their operator algebra structure and spectral synthesis properties.

Findings

Convolution maps are studied on the Haagerup tensor product of Fourier algebras.

The algebra $(A(G),\ast)$ is shown to be an operator algebra.

An unexpected set of spectral synthesis results is observed.

Abstract

We study the ranges of the maps of convolution $u\otimes v\mapsto u\ast v$ and a `twisted' convolution $u\otimes v\mapsto u\ast \check{v}$ ($\check{u}(s)=u(s^{-1})$) and on the Haagerup tensor product of a Fourier algebra of a compact group $A(G)$ with itself. We compare the results to result of factoring these maps through projective and operator projective tensor products. We notice that $(A(G),\ast)$ is an operator algebra and observe an unexpected set of spectral synthesis.