An application of Ramanujan graphs to C*-algebra tensor products

In a remarkable recent paper, Junge and Pisier (1995) prove that there are several distinct C*-norms on the tensor product B(H) x B(H), where B(H) is the C*-algebra of bounded linear operators on the usual Hilbert space H. To give a quantitative version of this result, they introduce the function lambda(n) = sup{\\u\\(max)/\\u\\(min): u a tensor with rank at most n in B(H) x B(H)}, and prove cn(1/8) less than or equal to lambda(n) less than or equal to n(1/2) for n > 2. In this note, we use Ramanujan graphs to get 1/2n(1/2) < lambda(n) for any n = q + 1, q a prime power. From this we deduce lim inf/pi-->infinity lambda(n)/root n greater than or equal to 1/2 root 3.