  Let $F_n$, $n\geq2$, be the free group with $n$ generators, denoted by
$U_1,U_2,...,U_n$. Let $C*(F_n)$ be the full $C^*$-algebra of $F_n$. Let
$\mathcal{X}$ be the vector subspace of the algebraic tensor product $C^*(F_n)
\otimes C^*(F_n)$, spanned by $1\otimes1,U_1\otimes1,...,U_n\otimes1,1\otimes
U_1,...,1\otimes U_n$. Let $|| \cdot ||_{\min}$ and $|| \cdot ||_{\max}$ be the
minimal and maximal $C^*$ tensor norms on $C^*(F_n) \otimes C^*(F_n)$, 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 $C^*(F_{2n})$ obtained by mapping $U_1\otimes1,...,1\otimes U_n$
into the $2n$ generators and the identity into the identity, we get a matrix
norm $|| \cdot ||_{C^*(F_{2n})}$ 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 $||X||_{\max} \leq ||X||_{C^*(F_{2n})} \leq
(N^2-N)^{1/2} ||X||_{\min}, X\in M_k(\mathbb{C})\otimes\mathcal{X}$.
