In Chapter 13 we introduced a method for analyzing mean differences on a single dependent variable between two or more independent groups while controlling the risk of making a Type I error at some specified α level. We explained the connection between the method and its name, analysis of variance, and showed how the method was a generalization of the independent group ttest.

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.

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 This mostly consists of unpublished work due to the late Uffe Haagerup.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

Often, variables in their original form do not lend themselves well to comparison with other variables or to certain types of analysis. In addition, often we may obtain greater insight by expressing a variable in a different form. For these and other reasons, in this chapter we discuss four different ways to re-express or transform variables: applying linear transformations, applying nonlinear transformations, recoding, and combining. We also describe how to use syntax files to manage your data analysis.

According to Equation 17.2, and as discussed in Chapter 16, holding relative humidity constant at a fixed value, each one-degree increase in temperature is associated with an estimated 0.88 more ice cream bars sold on average. This is true regardless of the value at which relative humidity is held constant. Likewise, holding temperature constant at a fixed value, each one-percentage increase in relative humidity is associated with an estimated 0.40 more ice cream bars sold on average regardless of the value at which temperature is held constant.

While random matrices give us the exact value of the constant C(n), it is natural to search for alternate deterministic constructions that show that C(n)<n. This chapter explores this direction. The central notion here is that of spectral gap. To prove the key estimate that C(n)<n, it suffices to produce sequences of n-tuples of unitary matrices exhibiting a certain kind of spectral gap. The notion of quantum expanders naturally enter the discussion here. Their existence can be derived from that of groups with property (T) admitting sufficiently many finite dimensional unitary representations. The notion of quantum spherical code that we introduce hereis a natural way to describe what is needed in the present context.

In Chapter 15, we were concerned with predicting the number of ice cream bars we could expect to sell at the beach from the daily highest temperature. Our goal was to predict that number as accurately as possible so that we would neither take too much ice cream to the beach or too little. Because of the linear shape depicted in the scatterplot and the strong relationship between number of ice cream sales and daily highest temperature (r = 0.887), we were able to construct a reasonably accurate simple linear regression equation for prediction with R2 = 0.78. But, we can try to do even better.

As noted in Chapter 1, the function of descriptive statistics is to describe data. A first step in this process is to explore how the collection of values for each variable is distributed across the array of values possible. Because our concern is with each variable taken separately, we say that we are exploring univariate distributions. Tools for examining such distributions, including tabular and graphical representations, are presented in this chapter along with Stata commands for implementing these tools.

Welcome to the study of statistics! It has been our experience that many students face the prospect of taking a course in statistics with a great deal of anxiety, apprehension, and even dread. They fear not having the extensive mathematical background that they assume is required, and they fear that the contents of such a course will be irrelevant to their work in their fields of concentration.

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.

This chapter starts with an overview of the complex interpolation method, for pairs of Banach spaces. Our main application here is when the pair is formed of the same space X with two equivalent norms. Fix an integer n. We consider a C*-algebra A and the space X formedof n-tuples in A equipped with two norms: the row-norm and the column-norm. In that case we prove a remarkable formula identifying the interpolated norm of parameter 1/2 (the midpoint of the interpolation scale). The latter formula involves the maximal tensor product of A with its complex conjugate. This is a preparation for the next chapter.

A theoretical probability model is a mathematical representation of a class of experiments having certain specified characteristics in common, and from which we may derive the probabilities of outcomes of any experiment in the class. The extent to which the probabilities derived from the theoretical model are correct for a particular experiment depends on the extent to which the particular experiment possesses the characteristics required by the model.

We study here the maps (defined on an operator space with values in a C*-algebra) that are bounded when "tensorized" with the identity of any other C*-algebra with respect to either the minimal or the maximal tensor product. More generally, we address here several natural questions inspired by category theory, related to injectivity and projectivity of morphisms.

One of Kirchberg’s conjecture that we emphasize here is whether the LLP implies the WEP. This actually reduces to the case of the full C* algebra C of the free group with countably infinitely many generators, which is the prototypical example with the LLP. The question is shown to be equivalent to a very simple inequality, involving the linear span of the unitary generators of C, that seems to be related to Grothendieck’s classicalinequality from Banach space theory. Various results are proved that tend to « almost prove » the conjecture, notably one by Tsirelson in which it would suffice to replace real scalars by complex ones to obtain the full conjecture.

We describe the minimal and maximal C*-tensor products and the states that they carry. We include a brief preliminary description of nuclear C*-algebras and we discuss the specific questions involving quotient C*-algebras.