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

Abstract

Let ${\mathbb{F}}_n$ be the free group of rank $n$ and let $⨁C^{\ast }({\mathbb{F}}_n)$ denote the direct sum of full group C${}^{\ast }$-algebras $C^{\ast }({\mathbb{F}}_n)$ of ${\mathbb{F}}_n$ $(1\le n<\infty$). We introduce a new comultiplication ${\Delta }_{\phi }$ on $⨁C^{\ast }({\mathbb{F}}_n)$ such that $(⨁C^{\ast }({\mathbb{F}}_n),{\Delta }_{\phi })$ is a non-cocommutative C${}^{\ast }$-bialgebra. With respect to ${\Delta }_{\phi }$, the tensor product $\pi {\otimes }_{\phi }{\pi }^{\prime }$ of any two representations $\pi$ and ${\pi }^{\prime }$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of several representations.

arXiv comment25 pages

References in corpus (1)

• A tensor product of representations of Cuntz algebras

Cited by in corpus (1)

• C${}^{\ast }$-bialgebra defined as the direct sum of UHF algebras