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.

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 This mostly consists of unpublished work due to the late Uffe Haagerup.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