Regular, quasi-regular and induced representations of infinite-dimensional groups
type: Item
instance of: scholarly article
title: Regular, quasi-regular and induced representations of infinite-dimensional groups
zbMATH Open document ID: 1394.22001
DOI: 10.4171/181
author: A. V. Kosyak
published in: EMS Tracts in Mathematics
publication date: 2018-06-13T00:00:00Z
review text: The main aim of this book is to start a systematic development of non-commutative harmonic analysis on infinite dimensional non-locally compact groups. In particular, the book generalizes and studies the notions of regular, quasi-regular and induced representations of infinite dimensional groups and develops an analogue of the orbit method in the case of infinite dimensional ``nilpotent'' groups. The book consists of twelve chapters, the bibliography and an index. Chapter 1 contains a lot of classical preliminary material on representations of locally compact groups which makes the book fairly self-contained. Chapter 2 studies irreducibility for the right regular representations of the ``nilpotent'' groups \(B_0^{\mathbb{N}}\) and \(B_0^{\mathbb{Z}}\) of infinite (in one or two directions, respectively) upper triangular matrices which have units on the main diagonal and only finitely many non-zero off-diagonal elements. An isomorphism criterion of such representations in case they are irreducible is given. Chapters 3 and 4 study quasi-regular representations of both \(B_0^{\mathbb{N}}\) and \(B_0^{\mathbb{Z}}\) and also of the ``solvable'' group \(Bor_0^{\mathbb{N}}\) of infinite in one direction upper triangular matrices. In Chapter 5 one finds a proof of the Dixmier commutation theorem for \(B_0^{\mathbb{N}}\) which asserts that the von Neumann algebras generated by the right regular and by the left regular representations of\(B_0^{\mathbb{N}}\) coincide. This allows the construction of the operator of modular conjugation and the modular operator used in Tomita-Takesaki theory. Chapter~6 determines when the von Neumann algebra generated by the right (resp. left) regular representations of \(B_0^{\mathbb{N}}\) and \(B_0^{\mathbb{Z}}\) is a factor and also determines the type of this factor. Chapter 7 studies induced representations. In particular, the classical construction, due to Mackey, of induced representations of locally compact groups is generalized to infin…
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 22-06
Mathematics Subject Classification ID: 22E66
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 6887448
zbMATH Keywords: matrix group
zbMATH Keywords: measure
zbMATH Keywords: current group
zbMATH Keywords: locally compact group
zbMATH Keywords: representation
zbMATH Keywords: irreducible representation
zbMATH Keywords: regular representation
zbMATH Keywords: quasi-regular representation
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: Hilbert space
zbMATH Keywords: tensor product
zbMATH Keywords: integral
zbMATH Keywords: induced representation
zbMATH Keywords: commutant
zbMATH Keywords: diffeomorphism
zbMATH Keywords: \(C^*\) algebra
zbMATH Keywords: Wiener space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/181
OpenAlex ID: W2802172491
description: scientific article; zbMATH DE number 6887448