Dye's theorem and Gleason's theorem for \(AW^\ast\)-algebras
type: Item
instance of: scholarly article
title: Dye's theorem and Gleason's theorem for \(AW^\ast\)-algebras
zbMATH Open document ID: 1320.46041
author: Jan Hamhalter
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1408.4597
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 81R15
zbMATH DE Number: 6362866
zbMATH Keywords: Dye's theorem
zbMATH Keywords: Gleason's theorem
zbMATH Keywords: \(AW^\ast\)-algebras
reviewed by: C. M. Edwards
MaRDI profile type: Publication
OpenAlex ID: W2964333404
arXiv ID: 1408.4597
cites work: Physical states on a \(C^*\)-algebra
cites work: Dimension functions and traces on C*-algebras
cites work: Complex Measures on Projections in Von Neumann Algebras
cites work: The Mackey-Gleason Problem
cites work: On Dye's theorem for Jordan operator algebras
cites work: The Mackey-Gleason Problem for Vector Measures on Projections in Von Neumann Algebras
cites work: A topological characterization of linearity for quasi-traces
cites work: Skew-symmetric functions on the sphere and quantum measures
cites work: Measures on projections and physical states
cites work: Abelian subalgebras and the Jordan structure of a von Neumann algebra
cites work: On the geometry of projections in certain operator algebras
cites work: Q3245484
cites work: Quasitraces on exact C*-algebras are traces
cites work: Quantum measure theory
cites work: Isomorphisms of ordered structures of Abelian \(C^*\)-subalgebras of \(C^*\)-algebras
cites work: STRUCTURE OF ASSOCIATIVE SUBALGEBRAS OF JORDAN OPERATOR ALGEBRAS
cites work: Active lattices determine \(\mathrm{AW}^\ast\)-algebras
cites work: Q3736359
cites work: Projections in Banach algebras
cites work: Q5723020
cites work: PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS
cites work: Measures on projections in \(W^ *\)-factors
cites work: Q3844299
cites work: Algebraic Theory of Continuous Geometries
cites work: Q3256710
cites work: Measures on Projections in W*-algebras of Type II1
cites work: Finitely Additive Measures on Projections in Finite W* -Algebras
DOI: 10.1016/J.JMAA.2014.09.040
review text: A famous \(C^*\)-algebraist, in his own defence, once remarked that every \(C^*\)-algebraist should be allowed to write one, but only one, paper on \(AW^\ast\)-algebras. Although the paper under review is an example of the genre, it nevertheless provides an interesting investigation into the properties of \(AW^\ast\)-algebras in which there has been a resurgence of interest in recent years.NEWLINENEWLINE In this paper, the author answers in the positive a question asked by \textit{C. Heunen} and \textit{M. L. Reyes} [J. Math. Anal. Appl. 416, No. 1, 289--313 (2014; Zbl 1310.46053)] of whether a function \(\phi\) from the complete lattice \({\mathcal P}(A)\) of projections in the \(AW^\ast\)-algebra \(A\), that has no type I\(_2\) direct summand, to \({\mathcal P}(B)\), where \(B\) is also an \(AW^\ast\)-algebra, that preserves arbitrary suprema and orthocomplements, automatically satisfies the condition that, for each pair \(p\) and \(q\) of elements in \({\mathcal P}(A)\), NEWLINE\[NEWLINE\phi((e_A- 2p) q(e_A- 2p))= (e_B- 2\phi(p)) \phi(q)(e_B- 2\phi(p)),NEWLINE\]NEWLINE where \(e_A\) and \(e_B\) are the units in \(A\) and \(B\), respectively. It then follows that \(\phi\) extends to a normal Jordan \(^*\)-homomorphism from \(A\) to \(B\). As a consequence of this result, he is able to generalize Dye's theorem to \(AW^\ast\)-algebras and show that the Mackey-Gleason-Bunce-Wright theorem also holds for a large class of \(AW^\ast\)-algebras. He then goes on to connect these results with his own work on order isomorphisms between the partially ordered sets of abelian sub-\(C^*\)-algebras of unital \(C^*\)-algebras.
description: scientific article; zbMATH DE number 6362866