Non-cocommutative C$^{*}$-bialgebra defined as the direct sum of free group C$^{*}$-algebras

TL;DR

This paper constructs a new non-cocommutative C*-bialgebra from the direct sum of free group C*-algebras, defining a novel tensor product of representations that is associative but not commutative.

Contribution

It introduces a new comultiplication on the direct sum of free group C*-algebras, creating a non-cocommutative C*-bialgebra and defining a novel tensor product of representations.

Findings

The tensor product operation is associative.

The tensor product operation is non-commutative.

Explicit formulas for tensor products of certain representations are provided.

Abstract

Let ${\Bbb F}_{n}$ be the free group of rank $n$ and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C$^{*}$-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n}$ $(1\leq n<\infty$). We introduce a new comultiplication $\Delta_{\varphi}$ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C$^{*}$-bialgebra. With respect to $\Delta_{\varphi}$, the tensor product $\pi\otimes_{\varphi}\pi'$ of any two representations $\pi$ and $\pi'$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of several representations.