Heisenberg relation for locally measurable operators
type: Item
instance of: scholarly article
title: Heisenberg relation for locally measurable operators
zbMATH Open document ID: 1409.46035
published in: Advances in Mathematics
publication date: 2018-08-29T00:00:00Z
review text: The authors address the question of which operators \(a, b\) on a Hilbert space \(H\) fulfill the Heisenberg relation in the sense that their commutator equals the unit: \([a,b]=1\). It is (well)known that such a relation is not possible if the two operators are bounded. Therefore, it is natural to consider the set of operators affiliated with a von Neumann algebra \(\mathcal{M}\). But since in general this set is not an algebra, the authors are naturally led to consider the smaller set of locally measurable operators \(LS(\mathcal{M})\) because \(LS(\mathcal{M})\) has the maximality property to include each unital \(^*\)-algebra of operators affiliated with \(\mathcal{M}\) and containing \(\mathcal{M}\) (where the algebraic structure is given by the so-called strong addition and strong multiplication). If \(\mathcal{M}\) is the Lebesgue space \(L^\infty(0,\infty)\) (or if \(\mathcal{M}\) is finite or has finite-dimensional centre), then \(LS(\mathcal{M})=S(\mathcal{M})\), where \(S(\mathcal{M})\) denotes the measurable operators affiliated with \(\mathcal{M}\) whereas, if \(\mathcal{M}\) is of type III with infinite-dimensional centre, then \(LS(\mathcal{M})\neq S(\mathcal{M})=\mathcal{M}\). (An operator \(x\) affiliated with \(\mathcal{M}\) is called measurable if \(\chi_{(N,\infty]}(|x|)\) is a finite projection for some \(N\), and it is called locally measurable if the centre of \(\mathcal{M}\) contains a sequence \((z_n)\) increasing to the unit such that \(z_n(H)\) is in \(\text{dom}(x)\) and \(xz_n\in S(\mathcal{M})\).) The first of the two main results recovers and improves, by elementary methods, known results, to wit: For \(a,b\in LS(\mathcal{M})\), \(\mathcal{M}\) a von Neumann algebra, each of the following three conditions implies \([a,b]\neq1\): \(\mathcal{M}\) is infinite or \(\mathcal{M}\) is a finite type I algebra or \(a\) is normal. The second main result is motivated by the known fact that, for \(a,x\in LS(\mathcal{M})\) (\(\mathcal{M}\) a von Ne…
reviewed by: Hermann Pfitzner
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46K05
zbMATH DE Number: 6929150
zbMATH Keywords: Heisenberg relation
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: locally measurable operators
author: A. F. Ber
author: Fedor Sukochev
author: Dmitriy Zanin
MaRDI profile type: Publication
full work available at URL: j.aim.2018.07.010
OpenAlex ID: W2884497250
cites work: On partially ordered real involutory algebras
cites work: DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS
cites work: Fuglede-Putnam theorem for locally measurable operators
cites work: Q3993197
cites work: Double operator integrals in a Hilbert space
cites work: Q4835025
cites work: Abstract characterization of \(EW^*\)-algebras
cites work: Q3927697
cites work: A Commutativity Theorem for Normal Operators
cites work: Q5571194
cites work: The Heisenberg relation -- mathematical formulations
cites work: Q3736359
cites work: Q4095424
cites work: Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations
cites work: Q4520849
cites work: On some mathematical aspects of the Heisenberg relation
cites work: Q5528957
cites work: Q5624220
cites work: A non-commutative extension of abstract integration
cites work: Operator synthesis. I: Synthetic sets, bilattices and tensor algebras
cites work: Operator synthesis II: Individual synthesis and linear operator equations
cites work: Q3857325
cites work: Theory of operator algebras I.
cites work: Q3891239
cites work: Über die Unbeschränktheit der Operatoren der Quantenmechanik
cites work: The Unboundedness of Quantum-Mechanical Matrices
cites work: On a Result of P. G. Dixon
Wikidata QID: Q129492026
DOI: 10.1016/J.AIM.2018.07.010
Recommended article: Q3099468
Recommended article: Q3622930
Recommended article: Q3633356
Recommended article: Localization operators on the Weyl-Heisenberg group
Recommended article: Heisenberg relations in the general case
Recommended article: Heisenberg-type inequalities for the Weinstein operator
Recommended article: Q4681783
Recommended article: The Heisenberg commutation relations, commuting squares and the Haar measure on locally compact quantum groups
Recommended article: ON HEISENBERG INEQUALITY
Recommended article: On some mathematical aspects of the Heisenberg relation
description: scientific article; zbMATH DE number 6929150