Tensor products of C*-algebras and operator spaces : the Connes-Kirchberg problem / by Gilles Pisier.

Includes bibliographical references and index.

"These notes are centered around the equivalence of two major open problems: one formulated by Connes (1976), about traces and ultraproducts of von Neumann algebras, the other one by Kirchberg (1993) about tensor products of C* algebras. This leads us to emphasize the notion of nuclear pair, that is a pair of C*-algebras admitting a unique tensor product. The main example is the pair (B,C) formed of the algebra B of bounded operators on Hilbert space and the full group C*-algebra C of any free group. This leads naturally to the weak expectation property (WEP) and the local lifting property (LLP), which we extensively study in connection with the more classical notions of nuclearity and exactness, or local reflexivity for C* algebras. We include two new characterizations of the WEP due to Haagerup but unpublished. We show that B fails the LLP and that the minimal tensor product of B with itself fails the WEP. Several properties of random unitary matrices and random permutations play a crucial role. We show the equivalence of the two main questions with a famous open one about Banach spaces and with Tsirelson's well known open problem in quantum information theory"-- Provided by publisher.