10 - Exactness and nuclearity

Summary

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.