A comparison between the max and min norms on C∗(Fn)⊗C∗(Fn).

Abstract. Let Fn, n > 2, be the free group on 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), spannedby 1 ­ 1,U1 ­ 1, . . . ,Un ­ 1, 1 ­ U1, . . . , 1 ­ Un. Let k · kmin and k · kmax 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, k 2 N. Identifying X with the subspace of C¤(F2n) obtained by mapping U1­ 1, . . . , 1­Un into the 2n generators and the identity into the identity, we get a matrix norm k · kC¤(F2n) which dominates the k · kmax norm on Mk(C)­X. In this paper we prove that, with N = 2n + 1 = dimX, we have kXkmax 6 kXkC¤(F2n) 6 (N2 − N)1/2kXkmin, X 2 Mk(C) ­ X