We review the basic theory of completely bounded (c.b.) and completely positive (c.p.) maps and prove the fundamental extension and factorization theorems about them.

This chapter is devoted to a proof that the Connes and Kirchberg questions are equivalent. We also formulate several equivalent variants of the conjectures, that somewhat complement the equivalence.

We describe the main properties of the bidual of a C*-algebra, that is also its enveloping von Neumann algebra. We define the tensor norms that naturally appear on a mixed pair formed of a C*-algebra and a von Neumann one, called the nor-norm and the bin-norm. This leads naturally to the notion of local reflexivity, which, in sharp contrast with the Banach space case, is not valid for all C*-algebras. Wegive explicit examples exhibiting that phenomenon, which is specific to the min-tensor product. Indeed, we show that the analogous defect disappears for the max-tensor product.

We prove here that the Connes and Kirchberg questions are equivalent to a different longstanding conjecture that circulated among Banach space theorists at least since the 1980’s if not sooner, namely the finite representability problem. The latter asks whether the predual of any von Neumann algebra is finitely representable in the trace class, or equivalent whether it embeds isometrically in an ultrapower (in the Banach space sense) of the trace class.

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.

This chapter presents Ozawa’s theorem that the minimal tensor product of B(H) with itself fails the WEP. The ingredients go through an investigation of the LLP for full crossed products of C*-algebras. Again the tools are either random matrices or deterministic examples related to property (T), but here it is crucial to work with unitary matrices associated to permutations.