Here we formulate the Connes embedding problem, whether any tracial probability space embeds in an ultraproduct of matricial ones. We also briefly describe the so-called hyperfinite factor R, with which one can reformulate the question as asking for an embedding in an ultrapower of R. Since the Connes problem is open even for the tracial probability spaces associated to discrete groups, this leads us to describe several related interesting classes of infinite groupssuch as residually finite, hyperlinear and sofic groups. We also discuss the so-called matrix models in terms of which the Connes problem can be naturally reformulated. Lastly, we give a quite transparent characterization of nuclear von Neumann algebras, which shows that there are very few of them.

In the remarkable paper where he proved the equivalence, Kirchberg studied more generally the pairs of C*-algebras(A,B) admitting only one C*-norm on their algebraic tensor product.We call such pairs "nuclear pairs''. A C*-algebra A istraditionally called nuclear if this holds for any C*-algebra B. Our exposition chooses as its cornerstone Kirchberg's theoremasserting the nuclearity of what is for us the "fundamental pair'', namely the pair (B,C)where B is the algebra B of bounded operators on Hilbert space and C isthe full group C*-algebra C of the free group with countably infinitely many generators. Our presentation leads us to highlight two properties of C*-algebras, the Weak Expectation Property (WEP) and the Local Lifting Property (LLP).

We prove a number of results on cb and cp maps in connection with operator space theory. In particular, we introduce the dual operator space.

Here we describe an example of group that is shown using Kazhdan’s property (T) to be such that its full C* algebrafails LLP, although the group is approximately linear (i.e. so-called hyperlinear). Since both amenable and free groups satisfy the latter LLP, it is not easy to produceexamples failing the LLP, and so far this is the only one.

This is a brief excursion into atopic that belongs to quantum information theory. We prove here that the Connes-Kirchberg problem is equivalent to a question raised by Tsirelson in connection with quantum mechanics. The question involves correlation matrices of various kinds that we discuss using operatorvalued measures in order to highlight the equivalence ofthe Tsirelson problem with the LLP of certain free product C*-algebras.

This chapter discusses the notions of exactness and nuclearity in connection with the completely bounded approximation property (CBAP) and the completely positive approximation property (CPAP).

This chapter generalizes and extends the development ofoperator-adapted wavelets (gamblets)and their resulting multiresolution decompositionsfrom Sobolev spaces to Banach spaces equipped with a quadraticnorm and a nonstandard dual pairing. The fundamental importance of the Schur complement is elucidated and the geometric nature of gamblets is presented from two views: one regarding basis transformations derived from the nesting, and the other the linear transformations associated with these basis transformations. A table of gamblet identities is presented.

This chapter reviews the correspondence between approximation theory and statistical inference.

This chapter presents, reviews, and summarizes fundamental concepts used throughout the book with the purpose of making it self-contained.

This chapter extends the presentation ofGaussian measures, cylinder measures, and fields fromSobolev spaces to Banach spaceswith quadratic energynorm. The relationship between weak distributions and cylinder measures is elucidated as is the relationship between Gaussian cylinder measures and Gaussian fields.

The introduction reviews, summarizes, and illustrates fundamental connections among Bayesian inference, numerical quadrature, Gausssian process regression, polyharmonic splines, information-based complexity, optimal recovery, and game theory that form the basis for the book. This is followed by describing a sample of the results derived from these interplays; including those in numerical homogenization, operator-adapted wavelets, fast solvers, and Gaussian process regression. It finishes with an outline of the structure of the book.

At the cost of some redundancy, to facilitate accessibility, the multiresolution decomposition and inversion of symmetric positive definite (SPD) matrices on finite-dimensional Euclidean space are developed in the Gamblet Transformand Decomposition framework.