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Published online by Cambridge University Press: 10 February 2020
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.
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