19 - Other proofs that C(n)<n: quantum expanders

Summary

In the paper where he formulated his famous conjecture that the LLP implies the WEP, Kirchberg actually conjectured that the converse also held. This was disproved shortly later on. This boils down to showing that B=B(H) fails the LLP, or equivalently that the pair (B,B) is not nuclear. We give a presentation of the construction that leads to this negative answer. The main point is in terms of a sequence of constants C(n) indexed by an integer n, and the negative answercan be derived rather quickly from the fact that C(n) < n for some n. We give various methods that prove this fact, including the most complete one that shows using random unitary matrices that C(n) is equal to twice the square root of n-1, and hence is <1 for all n>2. In passing this gives us a nice example showing that exactness is not stable under extensions, i.e. we can have an ideal I in some A such that both I and A/I are exact but A is not exact.Since the pair (B,B) is not nuclear, this means thatthere are two distinct C* norms on the tensor product of B with itself. We describe the more recent proof that there are infinitely many, and actually a whole continuum, of distinct such norms.