Lidé

# doc. RNDr. Martin Bohata, Ph.D.

## Všechny publikace

### Spectral order isomorphisms and AW*-factors

• Autoři:
• Publikace: Mathematische Nachrichten. 2022, 295(1), 6-21. ISSN 0025-584X.
• Rok: 2022
• DOI: 10.1002/mana.202000043
• Odkaz:
• Pracoviště:
• Anotace:
The paper deals with spectral order isomorphisms in the framework of AW*-algebras. We establish that every spectral order isomorphism between sets of all self-adjoint elements (or between sets of all effects, or between sets of all positive elements) in AW*-factors of Type I has a canonical form induced by a continuous function calculus and an isomorphism between projection lattices. In particular, this solves an open question about spectral order automorphisms of the set of all (bounded) self-adjoint operators on an infinite-dimensional Hilbert space. We also discuss spectral order isomorphisms preserving, in addition, orthogonality in both directions.

### Vigier's theorem for the spectral order and its applications

• Autoři:
• Publikace: Journal of Mathematical Analysis and Applications. 2019, 476(2), 801-810. ISSN 0022-247X.
• Rok: 2019
• DOI: 10.1016/j.jmaa.2019.04.016
• Odkaz:
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• Anotace:
The paper mainly deals with suprema and infima of self-adjoint operators in a von Neumann algebra M with respect to the spectral order. Let M-sa be the self-adjoint part of M and let <= be the spectral order on M-sa. We show that a decreasing net in (M-sa, <=) with a lower bound has the infimum equal to the strong operator limit. The similar statement is proved for an increasing net bounded above in (M-sa, <=) This version of Vigier's theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self-adjoint operators in terms of the strong operator limit and operator means. As an application of our results on suprema and infima, we study the order topology on M-sa, with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology.

### PREDUALS OF JBW*-TRIPLES ARE 1-PLICHKO SPACES

• DOI: 10.1093/qmath/hax057
• Odkaz:
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• Anotace:
We investigate the preduals of JBW*-triples from the point of view of Banach space theory. We show that the algebraic structure of a JBW*-triple M naturally yields a decomposition of its pre-dual M*, by showing that M* is a 1-Plichko space (that is, it admits a countably 1-norming Markushevich basis). In case M is sigma-finite, its predual M* is even weakly compactly generated. These results are a common roof for previous results on L-1-spaces, preduals of von Neumann algebras, and preduals of JBW*-algebras.

### Star Order and Topologies on von Neumann Algebras

• Autoři:
• Publikace: MEDITERRANEAN JOURNAL OF MATHEMATICS. 2018, 15(4), 1-14. ISSN 1660-5446.
• Rok: 2018
• DOI: 10.1007/s00009-018-1221-x
• Odkaz:
• Pracoviště:
• Anotace:
The goal of this paper is to study a topology generated by the star order on von Neumann algebras. In particular, it is proved that the order topology under investigation is finer than -strong* topology. On the other hand, we show that it is comparable with the norm topology if and only if the von Neumann algebra is finite-dimensional.

### Decompositions of preduals of JBW and JBW algebras

• DOI: 10.1016/j.jmaa.2016.08.031
• Odkaz:
• Pracoviště:
• Anotace:
We prove that the predual of any JBW*-algebra is a complex 1-Plichko space and the predual of any JBW-algebra is a real 1-Plichko space. I.e., any such space has a countably 1-norming Markushevich basis, or, equivalently, a commutative 1-projectional skeleton. This extends recent results of the authors who proved the same for preduals of von Neumann algebras and their self-adjoint parts. However, the more general setting of Jordan algebras turned to be much more complicated. We use in the proof a set-theoretical method of elementary submodels. As a byproduct we obtain a result on amalgamation of projectional skeletons. (C) 2016 Elsevier Inc. All rights reserved.

### On Markushevich bases in preduals of von Neumann algebras

• DOI: 10.1007/s11856-016-1365-y
• Odkaz:
• Pracoviště:
• Anotace:
We prove that the predual of any von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U. Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the selfadjoint part of the predual is 1-Plichko as well.

### Star order on operator and function algebras and its nonlinear preservers

• Autoři:
• Publikace: Linear and Multilinear Algebra. 2016, 64(12), 2519-2532. ISSN 0308-1087.
• Rok: 2016
• DOI: 10.1080/03081087.2016.1164661
• Odkaz:
• Pracoviště:
• Anotace:
The aim of this paper is to study the star order on operator and function algebras. It is shown that the infimum problem and the supremum problem on algebras of continuous functions have negative answer in general. Furthermore, we give a description of certain nonlinear star order isomorphisms between-algebras. Finally, we describe general star order isomorphisms on normal parts of matrix algebras and atomic von Neumann algebras.

### Symbióza kvantové teorie a funkcionální analýzy

• Autoři:
• Publikace: Československý časopis pro fyziku. 2016, 66(3), 153-156. ISSN 0009-0700.
• Rok: 2016
• Pracoviště:
• Anotace:
Jsou uvedeny některé zajímavé aspekty symbiózy mezi kvantovou teorií a funkcionální analýzou. V článku ukážeme, že některé hlubší problémy obou disciplín jsou překvapivě ekvivalentní.

### Star order on JBW algebras

• Autoři:
• Publikace: Journal of Mathematical Analysis and Applications. 2014, 2(417), 873-888. ISSN 0022-247X.
• Rok: 2014
• DOI: 10.1016/j.jmaa.2014.03.054
• Odkaz:
• Pracoviště:
• Anotace:
The star order is extended from associative algebras to non-associative Jordan Banach structures. After showing basic properties of this order, we discuss continuous (not necessarily linear) bijections preserving star order in the context of JBW algebras. In particular, we show that these maps between JBW factors of Type I\$_n\$, where \$nneq 2\$, have the form given by composition of Jordan isomorphism with functional calculus.

### Nonlinear maps on von Neumann algebras preserving the star order

• Autoři:
• Publikace: Linear and Multilinear Algebra. 2013, 61(7), 998-1009. ISSN 0308-1087.
• Rok: 2013
• DOI: 10.1080/03081087.2012.721363
• Odkaz:
• Pracoviště:
• Anotace:
Star order is defined on a C*-algebra in the following way: a {precedes above single-line equals sign} b if a*a = a*b and aa* = ba*. Let A{script} be a von Neumann algebra without Type I2 direct summand. Let A{script}n be the set of all normal elements of A{script}. Suppose that φ{symbol}: A{script}n → A{script}n is a continuous bijection that preserves the star order on A{script}n in both directions. Further, let there is a function f: ℂ → ℂ and an invertible central element c in A{script} such that φ{symbol}(λ1) = f(λ)c for all λ ∈ ℂ. We show that there is a unique Jordan *-isomorphism ψ: A{script} → A{script} such that Ramifications of this result as well as optimality of the assumptions are discussed

### Generalized Buneman Dispersion Relation in the Longitudinally Dominated Magnetic Field

• DOI: 10.5402/2011/896321
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• Anotace:
The generalized Buneman dispersion relation for two-component plasma is derived in the case of nonzero pressure of both plasma components and longitudinally dominated magnetic field. The derived relation is also valid for other field configurations mentioned in the paper. It can be useful in a variety of plasma systems, e.g. in the analyses of plasma jet penetrating into background plasma, in beam-plasma physics and in tests of various magnetohydro-dynamical (MHD) and hybrid numerical codes designed for the magnetized plasmas

### Operator Algebras and Quantum Structures

• Autoři:
• Publikace: Workshop 2011. Praha: České vysoké učení technické v Praze, 2011. pp. 1-3.
• Rok: 2011
• Pracoviště:
• Anotace:
The work deals with the star order on C*-algebras. The infinite C*-algebras are characterized in terms of the star order. Further, the infimum and supremum problem for the star order on function algebras is investigated.

### Star order on operator and function algebras

• Autoři:
• Publikace: Publicationes Mathematicae. 2011, 79(1-2), 211-229. ISSN 0033-3883.
• Rok: 2011
• DOI: 10.5486/PMD.2011.5006
• Odkaz:
• Pracoviště:
• Anotace:
The paper deals with the star order on proper *-algebras. Many results on the star order on matrix algebras and algebras of bounded operators acting on a Hilbert space are generalized to the C*-algebraic context. We characterize the star order on partial isometries in proper *-algebras in terms of their initial and final projections. As a corollary, we present a new characterization of infinite C*-algebras. Further, main results concern the infimum and supremum problem for the star order on a C*-algebra C(X) of all continuous complex-valued functions on a Hausdorff topological space X. We show that if X is locally connected or hyperstonean, then any upper bounded set in C(X) has an infimum and a supremum in the star order.

### Bell's Correlations and Spin Systems

• Autoři:
• Publikace: Foundations of Physics. 2010, 40(8), 1065-1075. ISSN 0015-9018.
• Rok: 2010
• DOI: 10.1007/s10701-009-9401-y
• Odkaz:
• Pracoviště:
• Anotace:
The structure of maximal violators of Bell's inequalities for Jordan algebras is investigated. It is proved that the spin factor V (2) is responsible for maximal values of Bell's correlations in a faithful state. In this situation maximally correlated subsystems must overlap in a nonassociative subalgebra. For operator commuting subalgebras it is shown that maximal violators have the structure of the spin systems and that the global state (faithful on local subalgebras) acts as the trace on local subalgebras.

### Bell's Inequalities and Pauli Matrices

• Autoři:
• Publikace: Abstr. Internat. Conf. Quantum Structures 2009. Bratislava: Slovak University of Technology in Bratislava, 2009. pp. 12. ISBN 978-80-227-3175-1.
• Rok: 2009
• Pracoviště:
• Anotace:
Bell's inequalities and their maximal violators are investigated.

### Determination of Deuteron Energy Distribution From Neutron Diagnostics in a Plasma-Focus Device

• DOI: 10.1109/TPS.2008.2005899
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• Anotace:
Fast neutrons from deuteron-deuteron fusion reactions were used for a study of fast deuterons in the PF-1000 plasma-focus device. The energy spectrum of neutrons was determined by the time-of-flight method using ten scintillation detectors positioned downstream, upstream, and side-on the experimental facility. Neutron energy-distribution functions enabled the determination of axial and radial components of energy of deuterons producing the fusion neutrons, as well as a rough evaluation of the total energy distribution of all fast deuterons in the pinch. It was found that the total deuteron energy-distribution function decreases with the deuteron energy more slowly than the tail of the Maxwellian distribution for 1-2-keV deuterons.

### Generalization of Bell's inequalities

• Autoři:
• Publikace: Workshop 09 CTU REPORTS. Praha: České vysoké učení technické v Praze, 2009. pp. 22-23. ISBN 978-80-01-04286-1.
• Rok: 2009
• Pracoviště:
• Anotace:
Bell's inequalities are generalized to *-algebras. The interesting structural consequences are investigated.

### Maximal violation of Bell's inequalities and Pauli spin matrices

• Autoři:
• Publikace: Journal of Mathematical Physics. 2009, 50(8), p082101. ISSN 0022-2488.
• Rok: 2009
• DOI: 10.1063/1.3190118
• Odkaz:
• Pracoviště:
• Anotace:
The paper deals with the structure of Bell's inequalities in the CHSH form. It is proved that Bell's inequalities are maximally violated for general *-algebras and faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived.

Za stránku zodpovídá: Ing. Mgr. Radovan Suk