Given a compact space K, we denote by P(K) the space of all Radon probability measures on K, equipped with the weak* topology inherited from C(K)*. For nonmetrizable compacta K even basic properties of P(K) spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject.
This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes–Kasparov conjecture in operator K-theory for these groups. In Part I we presented the Morita equivalence and the Connes–Kasparov morphism. In this part we shall compute the morphism using David Vogan’s description of the tempered dual. In fact we shall go further by giving a complete representation-theoretic description and parametrization, in Vogan’s terms, of the essential components of the tempered dual, which carry the K-theory of the tempered dual.
The purpose of this article is to give an exposition of a proof of the distributional form of the Whittaker Plancherel Theorem. The proof is an application of Harish-Chandra’s Plancherel Theorem for real reductive groups and its exposition can be used as an introduction to Harish-Chandra’s Plancherel Theorem. The paper follows the basic method in the author’s original approach in his second volume on real reductive groups. An error in the calculation of the Whittaker Transform of a Harish-Chandra wave packet is fixed using a result of Raphaël Beuzart-Plessis.
This is the first of two papers dedicated to the detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan’s approach to the classification of the tempered dual.
We define a Lax operator as a monic pseudodifferential operator L(∂) of order N ≥ 1, such that the Lax equations (frac{partial L(partial)}{partial t_k}=[(L^frac{k}{N}(partial))_+,L(partial)]) are consistent and non-zero for infinitely many positive integers k. Consistency of an equation means that its flow is defined by an evolutionary vector field. In the present paper we demonstrate that the traditional theory of the KP and the N-th KdV hierarchies holds for arbitrary scalar Lax operators.
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