An application of Ramanujan graphs to C∗-algebra tensor products O

In a remarkable recent paper, Junge and Pisier (1995) prove that there are several distinct C∗-norms on the tensor product B(H) ⊗ 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 and prove cn18 ≤ λ(n) ≤ n12 for n > 2. In this note, we use Ramanujan graphs to get 12n12 < λ(n) for any n = q + 1, q a prime power. From this we deduce lim infn→∞λ(n)n⩾123.