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 This mostly consists of unpublished work due to the late Uffe Haagerup.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

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.