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

Abstract

Let $F_n$, $n\ge 2$, be the free group with $n$ generators, denoted by $U_1,U_2,...,U_n$. 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 $1\otimes 1,U_1\otimes 1,...,U_n\otimes 1,1\otimes U_1,...,1\otimes U_n$. Let $||\cdot |{|}_{\min }$ and $||\cdot |{|}_{\max }$ 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 $M_k(\mathbb{C})\otimes \mathcal{X}$. Identifying $\mathcal{X}$ with the subspace of $C^{\ast }(F_{2n})$ obtained by mapping $U_1\otimes 1,...,1\otimes U_n$ into the $2n$ generators and the identity into the identity, we get a matrix norm $||\cdot |{|}_{C^{\ast }(F_{2n})}$ which dominates the $||\cdot |{|}_{\max }$ norm, on $M_k(\mathbb{C})\otimes \mathcal{X}$. In this paper we prove that, with $N=2n+1=\dim \mathcal{X}$, we have $||X|{|}_{\max }\le ||X|{|}_{C^{\ast }(F_{2n})}\le (N^2-N{)}^{1/2}||X|{|}_{\min },X\in M_k(\mathbb{C})\otimes \mathcal{X}$.

arXiv comment11 pages, AMS-LaTeX

References in corpus (1)

• A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms