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Davide Pastorello - One of the best experts on this subject based on the ideXlab platform.
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generalized complex spherical harmonics frame functions and Gleason Theorem
Annales Henri Poincaré, 2013Co-Authors: Valter Moretti, Davide PastorelloAbstract:Consider a finite dimensional complex Hilbert space \({\mathcal{H}}\), with \({dim(\mathcal{H}) \geq 3}\), define \({\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}\), and let \({\nu_\mathcal{H}}\) be the unique regular Borel positive measure invariant under the action of the unitary operators in \({\mathcal{H}}\), with \({\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}\). We prove that if a complex frame function \({f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}\) satisfies \({f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}\), then it verifies Gleason’s statement: there is a unique linear operator \({A: \mathcal{H} \to \mathcal{H}}\) such that \({f(u) = \langle u| A u\rangle}\) for every \({u \in \mathbb{S}(\mathcal{H}).\,A}\) is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.
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generalized complex spherical harmonics frame functions and Gleason Theorem
arXiv: Mathematical Physics, 2012Co-Authors: Valter Moretti, Davide PastorelloAbstract:Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $\nu_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\cH$, with $\nu_\cH(\bS(\cH))=1$. We prove that if a complex frame function $f : \bS(\cH)\to \bC$ satisfies $f \in \cL^2(\bS(\cH), \nu_\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \cH \to \cH$ such that $f(u) = $ for every $u \in \bS(\cH)$. $A$ is Hermitean when $f$ is real. No boundedness requirement is thus assumed on $f$ {\em a priori}.
Valter Moretti - One of the best experts on this subject based on the ideXlab platform.
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generalized complex spherical harmonics frame functions and Gleason Theorem
Annales Henri Poincaré, 2013Co-Authors: Valter Moretti, Davide PastorelloAbstract:Consider a finite dimensional complex Hilbert space \({\mathcal{H}}\), with \({dim(\mathcal{H}) \geq 3}\), define \({\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}\), and let \({\nu_\mathcal{H}}\) be the unique regular Borel positive measure invariant under the action of the unitary operators in \({\mathcal{H}}\), with \({\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}\). We prove that if a complex frame function \({f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}\) satisfies \({f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}\), then it verifies Gleason’s statement: there is a unique linear operator \({A: \mathcal{H} \to \mathcal{H}}\) such that \({f(u) = \langle u| A u\rangle}\) for every \({u \in \mathbb{S}(\mathcal{H}).\,A}\) is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.
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generalized complex spherical harmonics frame functions and Gleason Theorem
arXiv: Mathematical Physics, 2012Co-Authors: Valter Moretti, Davide PastorelloAbstract:Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $\nu_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary operators in $\cH$, with $\nu_\cH(\bS(\cH))=1$. We prove that if a complex frame function $f : \bS(\cH)\to \bC$ satisfies $f \in \cL^2(\bS(\cH), \nu_\cH)$, then it verifies Gleason's statement: There is a unique linear operator $A: \cH \to \cH$ such that $f(u) = $ for every $u \in \bS(\cH)$. $A$ is Hermitean when $f$ is real. No boundedness requirement is thus assumed on $f$ {\em a priori}.
Jan Hamhalter - One of the best experts on this subject based on the ideXlab platform.
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quantum measure theory
2003Co-Authors: Jan HamhalterAbstract:Preface. 1: Introduction. 2: Operator Algebras. 2.1. C*-Algebras. 2.2. Von Neumann Algebras. 2.3. Jordan Algebras And Ordered Structures. 3: Gleason Theorem. 3.1. Reduction To Three-Dimensional Space. 3.2. Regularity Of Frame Functions On R3. 3.3. Boundedness Of Frame Functions. 3.4. Historical Remarks And Comments. 4: Completeness Criteria. 4.1. Functiona1 Completeness Criteria. 4.2. Algebraic Completeness Criteria. 4.3. Measure Theoretic Completeness Criteria. 4.4. Historical Remarks And Comments. 5: Generalized Gleason Theorem. 5.1. The Mackey-Gleason Problem. 5.2. Reduction To Scalar Quasi-Functionals. 5.3. Linear Extensions Of Measures On Type In Algebras. 5.4. Linear Extensions Of Measures On Infinite Algebras. 5.5. Linear Extensions Of Measures On Finite Algebras. 5.6. Historical Remarks And Comments. 6: Basic Principles Of Quantum Measure Theory. 6.1. Boundedness Of Completely Additive Measures. 6.2. Yosida-Hewitt Decompositions Of Quantum Measures. 6.3. Convergence Theorems. 6.4. Historical Remarks And Comments. 7: Applications Of Gleason Theorem. 7.1. Multiform Gleason Theorem And Decoherence. 7.2. Velocity Maps And Derivations. 7.3. Approximate Hidden Variables. 7.4. Historical Remarks And Comments. 8: Orthomorphisms Of Projections. 8.1. Orthomorphisms Of Projection Lattices. 8.2. Countable Additivity Of *-Homomorphisms. 8.3. Historical Remarks And Comments. 9: Restrictions And Extensions Of States. 9.1. Restriction Properties Of Pure States. 9.2. Gleason Type Theorems For Quantum Logics. 9.3. Historical Remarks And Comments. 10: Jauch-Piron States. 10.1. Basic Properties Of Jauch States. 10.2. Nonsingularity Of Jauch-Piron States. 10.3. Countable Additivity Of States. 10.4. Historical Remarks And Comments. 11: Independence Of Quantum Systems. 11.1. Independence In Classical And Quantum Theory. 11.2. Independence Of C*-Algebras. 11.3. Independence Of Von Neumann Algebras. 11.4. Historical Remarks And Comments. Bibliography. Index.
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generalized Gleason Theorem
2003Co-Authors: Jan HamhalterAbstract:Gleason Theorem, which was a central subject of the previous chapters, says that any completely additive measure on the projection lattice of a Hilbert space extends to a linear functional on all bounded operators. The lattice of all projections on a Hilbert space H can be characterized among von Neumann projection lattices as being atomic and irreducible. Thus, Gleason Theorem covers only very special situation in this respect. Besides, it is important to describe all measures on projection lattices and not only completely additive ones. In this connection, a natural question arises of whether or not Gleason Theorem can be extended to finitely additive measures on projection lattices of general von Neumann algebra. This question was first posed by Mackey [224].
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applications of Gleason Theorem
2003Co-Authors: Jan HamhalterAbstract:Chapter 6 treated the consequences of the Generalized Gleason Theorem that are parallel to fundamental principles of standard measure theory and, simultaneously, they form the infrastructure of mathematical foundations of quantum theory. In this chapter we would primarily like to discuss those applications of Gleason Theorem that have direct physical meaning and provide solutions to problems originally posed in physics.
Mushtari D. - One of the best experts on this subject based on the ideXlab platform.
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The Gleason Theorem for the field of rational numbers and residue fields
2020Co-Authors: Mushtari D.Abstract:Charges μ taking values in a field F and defined on orthomodular partially ordered sets (logics) of all projectors in some finite-dimensional linear space over F are considered. In the cases where F is the field of rational numbers or a residue field, the Gleason representation μ(P) = tr(T μP), where T μ is a linear operator, is proved. © 1999 Kluwcr Academic/Plenum Publishers
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Gleason-type Theorem for linear spaces over the field of four elements
2020Co-Authors: Mushtari D.Abstract:We prove an analog of the famous Gleason Theorem for additive functions on the orthomodular poset of all projections defined on an n-dimensional linear space over the field consisting of four elements. An essential part of the proof consists in a computer calculation
Karimbergen Kudaybergenov - One of the best experts on this subject based on the ideXlab platform.
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2-Local derivations on von Neumann algebras
Positivity, 2014Co-Authors: Shavkat Ayupov, Karimbergen KudaybergenovAbstract:The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type \(III\) von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation.