Journal of Operator Theory

Volume 51, Issue 2, Spring 2004 pp. 245-253.

A comparison between the max and min norms on

$C^{\ast}(F_{n})\otimes C^{\ast}(F_{n})$

Authors: Florin Radulescu
Author institution: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA

Summary: Let

$F_{n}$ ,

$n\geq2$ , be the free group on

$n$ generators, denoted by

$U_{1},U_{2},\ldots,U_{n}$ . Let

$C^{\ast}(F_{n})$ be the full

$C^{\ast}$ -algebra of

$F_{n}$ . Let

$\mathcal{X}$ be the vector subspace of the algebraic tensor product

$C^{\ast}(F_{n})\otimes C^{\ast }( F_{n})$ , spanned by

$1\otimes1,U_{1}\otimes1,\ldots,U_{n}\otimes1,1\otimes U_{1},\ldots,1\otimes U_{n}$ . Let

$\Vert\,\cdot\,\Vert _{\min}$ and

$\Vert \,\cdot\,\Vert_{\max}$ be the minimal and maximal

$C^{\ast}$ tensor norms on

$C^{\ast}(F_{n})\otimes C^{\ast}(F_{n})$ , and use the same notation for the corresponding (matrix) norms induced on

$M_{k}(\mathbb{C})\otimes\mathcal{X}$ ,

$k\in\mathbb{N}$ . Identifying

$\mathcal{X}$ with the subspace of

$C^{\ast}(F_{2n})$ obtained by mapping

$U_{1}\otimes1,\ldots,1\otimes U_{n}$ into the

$2n$ generators and the identity into the identity, we get a matrix norm

$\Vert \,\cdot\,\Vert _{C^{\ast}(F_{2n}) }$ which dominates the

$\Vert \,\cdot\,\Vert_{\max}$ norm on

$M_{k}(\mathbb{C})\otimes\mathcal{X}$ . In this paper we prove that, with

$N=2n+1=\dim\mathcal{X}$ , we have

$\Vert X\Vert _{\max}\leq\Vert X\Vert _{C^{\ast}(F_{2n}) }\leq( N^{2}-N) ^{1/2}\Vert X\Vert _{\min},\quad X\in M_{k}( \mathbb{C}) \otimes\mathcal{X}.$

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