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# Journal of Operator Theory

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

A comparison between the max and min norms on $C^{\ast}(F_{n})\otimes C^{\ast}(F_{n})$

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