On the quantisation of spaces
type: Item
instance of: scholarly article
title: On the quantisation of spaces
zbMATH Open document ID: 1026.06018
DOI: 10.1016/S0022-4049(02)00139-1
author: Christopher J. Mulvey
author: Joan Wick Pelletier
published in: Journal of Pure and Applied Algebra
publication date: 2002-12-02T00:00:00Z
review text: Central to the theory of locales has been the connection between the notions of a point of a locale and that of a spatial locale. As the theory of quantales is on one level an attempt to generalize the theory of locales [see the reviewer's book, Quantales and their applications. Pitman Research Notes, Vol. 234. Harlow: Longman Scientific and Technical (1990; Zbl 0703.06007)], it is natural to try to develop a theory of points and spatiality in the context of quantales. This article is an extension of the fundamental work of the authors [``On the quantisation of points'', J. Pure Appl. Algebra 159, 231-295 (2001; Zbl 0983.18007)]. The setting is that of involutive unital quantales and one of the main goals is to indicate how the concepts of point and space for quantales provide the appropriate framework for looking at the spectral theory of \(C^*\)-algebras, one of the original motivations of quantale theory. After a useful review of concepts including those of involutive, Gelfand, Hilbert and von Neumann quantales from their earlier papers, they develop the theory of discrete von Neumann quantales as a prelude to defining an involutive unital quantale to be spatial provided that it admits an algebraically strong right embedding into a discrete von Neumann quantale. They proceed to prove many results including a characterization of such spatial quantales as well as the desired result that an involutive unital quantale is spatial iff it has enough points. They also show that the quantalic notion of spatiality agrees with that of spatiality for locales, when the latter are viewed as involutive, unital quantales. And finally it is shown that things work as expected for the case of the spectrum, \(\text{Max }A\), of a \(C^*\)-algebra \(A\).
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 18F99
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 1839072
zbMATH Keywords: points of quantales
zbMATH Keywords: involutive unital quantales
zbMATH Keywords: spectral theory of \(C^*\)-algebras
zbMATH Keywords: discrete von Neumann quantales
zbMATH Keywords: spatial quantales
zbMATH Keywords: spatiality
zbMATH Keywords: locales
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
cites work: Left ideal structure of \(C^*\)-algebras
cites work: The spectral theory of commutative C∗-algebras: The constructive spectrum
cites work: The spectral theory of commutative C∗-algebras: The constructive Gelfand-Mazur theorem
cites work: A globalisation of the Gelfand duality theorem
cites work: Quantales and their sheaves
cites work: An essay on noncommutative topology
cites work: Quantales and C∗-Algebras
cites work: Q5531996
cites work: Q5605136
cites work: Spatial quantales
cites work: Q3771067
cites work: Q5182330
cites work: Q3137162
cites work: On the quantisation of points
cites work: Von Neumann algebras and Hilbert quantales
cites work: Simple involutive quantales
cites work: Q3831512
cites work: Q5624220
full work available at URL: s0022-4049(02)00139-1
OpenAlex ID: W2148544922
Recommended article: Q4508558
Recommended article: Prime elements of non-integral quantales and their applications.
Recommended article: Non-commutative topology and quantales
Recommended article: On the quantisation of points
Recommended article: On quantales and spectra of C*-algebras
Recommended article: An essay on noncommutative topology
Recommended article: Quantales and C∗-Algebras
Recommended article: Simple involutive quantales
Recommended article: Q5755143
Recommended article: Q3137162
description: scientific article; zbMATH DE number 1839072