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 This mostly consists of unpublished work due to the late Uffe Haagerup.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.

Here we formulate the Connes embedding problem, whether any tracial probability space embeds in an ultraproduct of matricial ones. We also briefly describe the so-called hyperfinite factor R, with which one can reformulate the question as asking for an embedding in an ultrapower of R. Since the Connes problem is open even for the tracial probability spaces associated to discrete groups, this leads us to describe several related interesting classes of infinite groupssuch as residually finite, hyperlinear and sofic groups. We also discuss the so-called matrix models in terms of which the Connes problem can be naturally reformulated. Lastly, we give a quite transparent characterization of nuclear von Neumann algebras, which shows that there are very few of them.

In the remarkable paper where he proved the equivalence, Kirchberg studied more generally the pairs of C*-algebras(A,B) admitting only one C*-norm on their algebraic tensor product.We call such pairs "nuclear pairs''. A C*-algebra A istraditionally called nuclear if this holds for any C*-algebra B. Our exposition chooses as its cornerstone Kirchberg's theoremasserting the nuclearity of what is for us the "fundamental pair'', namely the pair (B,C)where B is the algebra B of bounded operators on Hilbert space and C isthe full group C*-algebra C of the free group with countably infinitely many generators. Our presentation leads us to highlight two properties of C*-algebras, the Weak Expectation Property (WEP) and the Local Lifting Property (LLP).