We present some fundamental aspects of the representation theory of reductive p-adic groups, mapping out some recent developments, including the ABPS conjecture and the description of the structure of the reduced C*-algebras of reductive groups. The exposition proceeds fairly rapidly, but is essentially self-contained.

We start by introducing C*-algebras associated with locally compact groups. Next, the theory of Hilbert modules, C*-correspondences, crossed-product algebras, and Morita equivalence are discussed. This is followed with applications to Mackey’s theory of induction of representations and Howe’s theory of theta correspondence. Next, K-theory and (equivariant) KK-theory are introduced. Connections between isolated points in the unitary dual and generators of the K-theory of C*-algebras of liminal groups are discussed.

We start by describing the fundamental theory of representation theory of a semisimple Lie group. This is followed by a classification of almost all irreducible tempered representations of a connected, semisimple, linear Lie group. We apply this classification to describe the structure of the reduced group C*-algebras of such groups.

This chapter is an introduction to the geometric perspective of tempered representations of connected reductive Lie groups via Dirac induction and the Connes–Kasparov isomorphism. We start with discussing Dirac operators in spin geometry. We then recall some classical Dirac operators on homogeneous spaces arising in representation theory. Next, we introduce the theory of higher index to be used as a language to formulate the Connes–Kasparov conjecture. In Section 3.4, we introduce Dirac induction, the Connes–Kasparov conjecture, and survey a couple of ways for proving it. The final section is devoted to the harmonic-analytic approach to the Connes–Kasparov isomorphism illustrated by the example of connected semisimple groups.

Chapter 6 opens with extensions of martingale theory in two directions: to σ-finite measures and to random variables with values in a Banach space. In §6.2 I prove Burkholder’s Inequality for martingales with values in a Hilbert space. The derivation that I give is essentially the same as Burkholder’s second proof, the one that gives optimal constants. Finally, the results in §6.1 are used in §6.3 to derive Birkhoff’s Individual Ergodic Theorem and a couple of its applications.

We introduce here basic concepts concerning equivalence relations, morphisms, and reductions, in general and in the Borel category.

Section 7.1 provides a brief introduction to the theory of martingales with a continuous parameter. As anyone at all familiar with the topic knows, anything approaching a full account of this theory requires much more space than a book like this can provide. Thus, I deal with only its most rudimentary aspects, which, fortunately, are sufficient for the applications to Brownian motion that I have in mind. Namely, in §7.2 I first discuss the intimate relationship between continuous martingales and Brownian motion (Lévy’s martingale characterization of Brownian motion), then derive the simplest (and perhaps most widely applied) case of the Doob–Meyer Decomposition Theory, and finally show what Burkholder’s Inequality looks like for continuous martingales. In the concluding section, §7.3, the results in §7.1 and §7.2 are applied to derive the Reflection Principle for Brownian motion.

This chapter is devoted to the study of infinitely divisible laws. It begins in §3.1 with a few refinements (especially the Lévy Continuity Theorem) of the Fourier techniques introduced in §2.3. These play a role in §3.2, where the Lévy–Khinchine formula is first derived and then applied to the analysis of stable laws.

Because they are not needed earlier, conditional expectations do not appear until Chapter 5. The advantage gained by this postponement is that, by the time I introduce them, I have an ample supply of examples to which conditioning can be applied; the disadvantage is that, with considerable justice, many probabilists feel that one is not doing probability theory until one is conditioning. Be that as it may, Kolmogorov’s definition is given in §5.1 and is shown to extend naturally to both σ-finite measure spaces and random variables with values in a Banach space. Section 5.2 presents Doob’s basic theory of real-valued, discrete parameter martingales: Doob’s Inequality, his Stopping Time Theorem, and his Martingale Convergence Theorem. In the last part of §5.2, I introduce reversed martingales and apply them to DeFinetti’s theory of exchangeable random variables.