On continuity of measurable group representations and homomorphisms
type: Item
instance of: scholarly article
title: On continuity of measurable group representations and homomorphisms
zbMATH Open document ID: 1290.22002
published in: Studia Mathematica
publication date: 2012-10-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0999
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 6090474
zbMATH Keywords: automatic continuity
zbMATH Keywords: Polish groups
zbMATH Keywords: group representations
zbMATH Keywords: homomorphisms of groups
zbMATH Keywords: non-measurable unions
zbMATH Keywords: unions of null sets
zbMATH Keywords: consistency of ZFC axioms and set theory
author: Yulia Kuznetsova
MaRDI profile type: Publication
OpenAlex ID: W2963215422
arXiv ID: 1010.0999
DOI: 10.4064/SM210-3-1
review text: The author is concerned with non-separable Hilbert space so chooses to investigate consistency with related axiomatic systems. \textit{K. Gödel} [The consistency of the continuum hypothesis. Princeton, N. J.: Princeton University Press (1940; Zbl 0061.00902)] showed that the continuum hypothesis is consistent with the axioms of set theory. The independence of the continuum hypothesis from ZFC (Zermelo-Fraenkel, axiom of choice) and of the independence of the axiom of choice from the remaining ZFC axioms follows from \textit{K. Gödel} [Am. Math. Mon. 54, 515--525 (1948; Zbl 0038.03003)] and \textit{P. J. Cohen} [Proc. Natl. Acad. Sci. USA 50, 1143--1148 (1964; Zbl 0192.04401)]. Both of these results assume that the ZFC axioms themselves do not contain a contradiction. The author also deals with Martin's axiom \textit{D. A. Martin} and \textit{R. M. Solovay} [Ann. Math. Logic 2, 143--178 (1970; Zbl 0222.02075)] which states that all cardinals less than \(c\), the cardinality of the continuum, behave like \(\aleph_{0}\); cf. \textit{D. H. Fremlin} [Manuscr. Math. 33, 387--405 (1981; Zbl 0459.28010)].NEWLINENEWLINELet \(G\) denote a locally compact group. \(U\) a unitary representation on a Hilbert space \(H\), \(\mathcal{L}(H)\) the space of bounded linear operators on \(H\) with the weak operator topology. Measurability is understood here in the sense of belonging to the \(\sigma\)-algebra generated by the family of Borel sets. NEWLINENEWLINEIn the first part of the article the author proves that a Haar measurable map from \(G\) to \(\mathcal{L}(H)\) is weakly (equivalently strongly) continuous. This is clear for separable \(H\) as it concerns mappings to complex-valued functions with no complications because of size. The proof for locally compact groups is based on the result for Polish spaces (cf. \textit{S. Banach} [Théorie des opérations linéaires. Warszawa, Mathematisches Seminar der Univ. Warschau (1932; JFM 58.0420.01)], who proved that measurable maps between …
reviewed by: Aubrey Wulfsohn
description: scientific article; zbMATH DE number 6090474