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

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 $Δ_φ$ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,Δ_φ)$ is a non-cocommutative C$^{*}$-bialgebra. With respect to $Δ_φ$, the tensor product $π\otimes_φπ'$ of any two representations $π$ and $π'$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of several representations.

25 pages