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

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

2010-11-28

arxiv.org/abs/1011.6034v1

Mathematics

Operator Algebras

20 pages

Scientific paper

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, and $C^{*}({\Bbb F}_{\infty})$ is a comodule-C$^{*}$-algebra of $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$. With respect to $\Delta_{\varphi}$, tensor product formulas of several representations of ${\Bbb F}_{n}$'s are computed. From these results, a similarity between $C^{*}({\Bbb F}_{n})$'s and Cuntz algebras are discussed.

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