A Comparison between the Max and Min Norms

Abstract. Let Fn, n ≥ 2, be the free group with n generators, denoted by U1, U2, . . . , Un. Let C ∗ (Fn) be the full C ∗-algebra of Fn. Let X be the vector subspace of the algebraic tensor product C∗ (Fn) ⊗ C∗ (Fn), spanned by 1 ⊗ 1, U1 ⊗ 1, . . . , Un ⊗ 1, 1 ⊗ U1, . . . , 1⊗Un. Let ‖ · ‖min and ‖ · ‖max be the minimal and maximal C∗ tensor norms on C∗ (Fn)⊗C (Fn), and use the same notation for the corresponding (matrix) norms induced on Mk (C)⊗X . Identifying X with the subspace of C∗ (F2n) obtained by mapping U1⊗1, . . . , 1⊗Un into the 2n generators and the identity into the identity, we get a matrix norm ‖ · ‖C∗(F2n) which dominates the ‖ · ‖max norm, on Mk (C)⊗X . In this paper we prove that, with N = 2n+1 = dimX , we have