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.

This chapter is an excursion into what could be called the local theoryof operator spaces. Here the main interest is on finite dimensional operator spaces and the degree of isomorphism of the various spaces is estimated using the c.b. analogue of the Banach-Mazur distance from Banach space theory. The main result is that the metric space formed of all the n-dimensional operator spaces equipped with the latter cb-distance is non separable for any n>2. This is in sharp contrast with the Banach space analogue which is a compact metric space.

We prove Kirchberg's theorem asserting that the fundamental pair (B,C) is nuclear where B is the algebra B of bounded operators on Hilbert space and C isthe full group C*-algebraof the free group with countably infinitely many generators. We then say that aC*-algebra A has the WEP (resp. LLP)if the pair (A,C) (resp. (A,B)) is nuclear. The generalized form of Kirchberg's theorem is then that any pair formed of a C*-algebra with the WEP and one with the LLP is nuclear. We show that the WEP of a C*-algebra A is equivalent to a certain extension property for maps on A with values in a von Neumann algebra, from which the term weak expectation is derived. In turn the LLP of A is equivalent to a certain local lifting property for maps on A with values in a quotient C*-algebra. We introduce the class of C*-algebras, called QWEP, that are quotients of C*-algebras with the WEP. One can also define analogues of the WEP and the LLP for linear maps between C*-algebras. Several properties can be generalized to this more general setting.

This chapter is devoted to the proof of two new characterizations of the WEP. This mostly consists of unpublished work due to the late Uffe Haagerup. Basically, the main point is as follows: consider an inclusion of a C*-algebra A into another (larger) one B. We wish to understand when there is a contractive projection from the bidual of B onto the bidual of A. From work presented earlier, we know that this holds if and only if the inclusion from A to B remains an inclusion if we tensorize it with any auxiliary C*-algebra C for the maximal tensor product. The main theorem of this chapter shows that actually a much weaker property suffices: it is enough to take for C the complex conjugate of A and we may restrict to « positive definite » tensors. The main case of interest is when B=B(H), in which case the property in question holds iff A has the WEP. Among the corollaries, one can prove that a von Neumann subalgebra of B(H) is injective as soon as there is a c.b. projection from B(H) onto it.

While random matrices give us the exact value of the constant C(n), it is natural to search for alternate deterministic constructions that show that C(n)<n. This chapter explores this direction. The central notion here is that of spectral gap. To prove the key estimate that C(n)<n, it suffices to produce sequences of n-tuples of unitary matrices exhibiting a certain kind of spectral gap. The notion of quantum expanders naturally enter the discussion here. Their existence can be derived from that of groups with property (T) admitting sufficiently many finite dimensional unitary representations. The notion of quantum spherical code that we introduce hereis a natural way to describe what is needed in the present context.

This chapter is a preparation for the formulation of the Connes embedding problem. We introduce tracial probability spaces (that is von Neumann algebras equipped with faithful, normaland normalized traces) and the so-called non-commutative L1 and L2 spaces associated to them.

The main examples that we describe are derived either from discrete groups or from semi-circular and circular systems, which are the analogues of Gaussian random variables in free probability. Wethen define ultraproducts of tracial probability spaces. This leads us to an important criterion for factorization of linear maps through B(H). We include a characterization of injectivity in terms of hypertraces, and we introduce the factorization property for discrete groups.

This chapter starts with an overview of the complex interpolation method, for pairs of Banach spaces. Our main application here is when the pair is formed of the same space X with two equivalent norms. Fix an integer n. We consider a C*-algebra A and the space X formedof n-tuples in A equipped with two norms: the row-norm and the column-norm. In that case we prove a remarkable formula identifying the interpolated norm of parameter 1/2 (the midpoint of the interpolation scale). The latter formula involves the maximal tensor product of A with its complex conjugate. This is a preparation for the next chapter.

We study here the maps (defined on an operator space with values in a C*-algebra) that are bounded when "tensorized" with the identity of any other C*-algebra with respect to either the minimal or the maximal tensor product. More generally, we address here several natural questions inspired by category theory, related to injectivity and projectivity of morphisms.

One of Kirchberg’s conjecture that we emphasize here is whether the LLP implies the WEP. This actually reduces to the case of the full C* algebra C of the free group with countably infinitely many generators, which is the prototypical example with the LLP. The question is shown to be equivalent to a very simple inequality, involving the linear span of the unitary generators of C, that seems to be related to Grothendieck’s classicalinequality from Banach space theory. Various results are proved that tend to « almost prove » the conjecture, notably one by Tsirelson in which it would suffice to replace real scalars by complex ones to obtain the full conjecture.

We describe the minimal and maximal C*-tensor products and the states that they carry. We include a brief preliminary description of nuclear C*-algebras and we discuss the specific questions involving quotient C*-algebras.

We review here the extension properties that are equivalent to the WEP. We take special care to clarify this topic because of some obscurity that we detected in Kirchberg’s original paper on this same topic. We also consider parallel lifting properties that express the LLP.

This chapter develops in great detail the theory of decomposable maps, that is maps that are linear combinations of c.p. maps. We make extensive use of the dec-norm due to Haagerup. The treatment we give for this topic, in connection with the maximal tensor product, seems new in book form.

This chapter recapitulates some of the main open problems that arise in connection with the main topics of these notes.

Here we study multiplicative domains of c.p. maps, that is subalgebras on which their restriction is a self adjoint preserving homomorphism. We also consider the same notion for Jordan morphisms.

Here we describe the C*-algebras, full (or maximal) and reduced, associated to a discrete group and we describe the known basic facts about multipliers acting on them. We present the basic characterizations of amenable groups in terms of their associated C*-algebras. We make frequent use in the sequelof the Fell's absorption principle, which is described here.