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prof. RNDr. Jan Hamhalter, CSc.

Všechny publikace

Continuous Preservers of Range Orthogonality between C*-Algebras

  • DOI: 10.1134/S1995080222100146
  • Odkaz: https://doi.org/10.1134/S1995080222100146
  • Pracoviště: Katedra matematiky
  • Anotace:
    We introduce new concept of algebraic *-orthogonality for maps between C*-algebras. We show that in case of bounded linear maps the algebraic*-orthogonality is equivalent to the property of preserving range orthogonality. Since algebraic orthogonality passes easily to the second dual, using spectral theory of von Neumann algebras we show canonical form of range orthogonality preservers in a relative short and lucid way.

Structure of preservers of range orthogonality on *-rings and C*-algebras

  • DOI: 10.1016/j.laa.2022.02.025
  • Odkaz: https://doi.org/10.1016/j.laa.2022.02.025
  • Pracoviště: Katedra matematiky
  • Anotace:
    The topic of this paper lies between algebraic theory of *-rings and *-algebras on one side, and analytic theory of C*-algebras on the other side. A map theta : A -> B between unital *-rings is called range orthogonal isomorphism if it is bijective and preserves range orthogonality in both directions. We show that any additive (resp. linear) range orthogonal isomorphism is canonical, that is, it is a *-isomorphism followed by multiplication from the right by an invertible element, provided that Ais generated by projections as a *-ring. In case of general * rings and *-algebras we show that direct summands generated by projections are well behaved with respect to range orthogonal morphisms. In particular, we show that additive range orthogonality isomorphisms are canonical on proper nonabelian parts of Baer *-algebras. We apply algebraic results to matrix C*-algebras to show that any range orthogonal isomorphisms between them is canonical. The same holds for C*-algebras having proper nonabelian part generated by projections. (C) 2022 Elsevier Inc. All rights reserved.

Symmetries of C ∗-algebras and Jordan Morphisms

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: Operator and Norm Inequalities and Related Topics. Cham: Birkhäuser, 2022. p. 673-705. ISSN 2297-0215. ISBN 978-3-031-02103-9.
  • Rok: 2022
  • DOI: 10.1007/978-3-031-02104-6_20
  • Odkaz: https://doi.org/10.1007/978-3-031-02104-6_20
  • Pracoviště: Katedra matematiky
  • Anotace:
    They are many faces of C ∗-algebras whose symmetries encode important aspects of their structures. We show that in surprisingly different situations these symmetries are implemented by Jordan *-isomorphisms and lead to full Jordan invariants. In this respect we study the following structures: 1. One dimensional projections in a Hilbert space with transition probability and orthogonality relation (Wigner type theorems). 2. Projection lattices of von Neumann algebras and AW∗-algebras (Dye type theorems) 3. Abelian C∗-subalgebras with set theoretic inclusion (Bohrification program in quantum theory) 4. Measures on state spaces endowed with the Choquet order.

Dye's theorem for tripotents in von Neumann algebras and JBW*-triples

  • DOI: 10.1007/s43037-021-00134-w
  • Odkaz: https://doi.org/10.1007/s43037-021-00134-w
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study morphisms of the generalized quantum logic of tripotents in JBW*-triples and von Neumann algebras. Especially, we establish a generalization of celebrated Dye's theorem on orthoisomorphisms between von Neumann lattices to this new context. We show the existence of a one-to-one correspondence between the following maps: (1) quantum logic morphisms between the posets of tripotents preserving reflection u -> -u (2) maps between triples that preserve tripotents and are real linear on sets of elements with bounded range tripotents. In a more general description we show that quantum logic morphisms on structure of tripotents are given by a family of Jordan *-homomorphisms on Peirce 2-subspaces. By examples we demonstrate optimality of the results. Besides we show that the set of partial isometrics with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.

GROTHENDIECK'S INEQUALITIES FOR JB*-TRIPLES: PROOF OF THE BARTON FRIEDMAN CONJECTURE

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Kalenda, O.F.K., Peralta, Antonio M., Pfitzner, H.
  • Publikace: Transactions of the American Mathematical Society. 2021, 374(2), 1327-1350. ISSN 0002-9947.
  • Rok: 2021
  • DOI: 10.1090/tran/8227
  • Odkaz: https://doi.org/10.1090/tran/8227
  • Pracoviště: Katedra matematiky
  • Anotace:
    We prove that, given a constant K > 2 and a bounded linear operator T from a JB*-triple E into a complex Hilbert space H, there exists a norm-one functional psi epsilon E* satisfying

Jordan Invariants of Von Neumann Algebras Given by Abelian Subalgebras and Choquet Order on State Spaces

  • Autoři: Turilova, E., prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: International Journal of Theoretical Physics. 2021, 60(2), 597-607. ISSN 0020-7748.
  • Rok: 2021
  • DOI: 10.1007/s10773-019-04157-w
  • Odkaz: https://doi.org/10.1007/s10773-019-04157-w
  • Pracoviště: Katedra matematiky
  • Anotace:
    New complete invariants for Jordan parts of von Neumann algebras are presented. We shall prove that the poset of all finite dimensional abelian von Neumann subalgebras ordered by set theoretic inclusion is a complete Jordan invariant for von Neumann algebras. On the other hand, we exhibit an example showing that not any order isomorphism on this structure is derived from a Jordan isomorphism. We apply our results to the Choquet order of orthogonal measures on state spaces of von Neumann algebras. Among others we show that the poset of decompositions of a fixed faithful normal state on a von Neumann algebra endowed with the Choquet order is a complete Jordan invariant for sigma-finite von Neumann algebras.

Lyapunov Convexity Theorem for von Neumann Algebras and Jordan Operator Structures

  • DOI: 10.1007/s00009-020-01624-1
  • Odkaz: https://doi.org/10.1007/s00009-020-01624-1
  • Pracoviště: Katedra matematiky
  • Anotace:
    We establish Lyapunov type theorems on automatic convexity of various affine transformations of the set of extreme points of important convex sets (closed unit ball, positive part of the closed unit ball, state space) appearing in the theory of von Neumann algebras and more general operator structures. Among others, we have shown that every bounded finitely additive measure mu :P(M)-> X, where P(M) is a projection lattice of a von Neumann algebra M with no sigma -finite direct summand, and X is a normed space with weak separable dual, has a convex range. Similar result is obtained for non sigma -finite JW factor. Further results along this line are proved for weak* continuous countably dimensional affine maps on closed unit balls of nonatomic JBW triples and on positive parts of nonatomic von Neumann algebras and JBW algebras.

Spectral Order for Jordan Triples

  • Autoři: Turilova, Ekaterina A., prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Proceedings of the Steklov Institute of Mathematics. 2021, 313(1), 258-262. ISSN 0081-5438.
  • Rok: 2021
  • DOI: 10.1134/S0081543821020231
  • Odkaz: https://doi.org/10.1134/S0081543821020231
  • Pracoviště: Katedra matematiky
  • Anotace:
    We initiate the study of the spectral order on Jordan triples. The order given on the tripotents is extended to the spectral order on the triples. We show that Jordan triples equipped with the spectral order are not lattices but preserve the Olson "momentum" characteristic.

Star Order and Partial Isometries in C*-Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: Lobachevskii Journal of Mathematics. 2021, 42(10), 2325-2332. ISSN 1995-0802.
  • Rok: 2021
  • DOI: 10.1134/S1995080221100085
  • Odkaz: https://doi.org/10.1134/S1995080221100085
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study the poset of partial isometries in C *-algebras endowed with the *-order and *-orthogonality. We show that this structure is a complete Jordan invariant for AW *-algebras. We prove that partial isometries in von Neumann algebras form a lower semilattice. The structures of partial isometries and projections are compared.

Completeness of Inner Product Spaces Associated with Functional on Jordan Structures

  • DOI: 10.1134/S1995080220040277
  • Odkaz: https://doi.org/10.1134/S1995080220040277
  • Pracoviště: Katedra matematiky
  • Anotace:
    We show that a normal functional varphi on a JBW* triple induces, via Gelfand–Naimark–Segal like construction, a complete inner product space if and only if varphi is a finite convex combination of extreme points from the predual. Application of this result to von Neumann algebras is shown.

Determinacy of Functionals and Lyapunov Theorem for Jordan Triple Structures and von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: Lobachevskii Journal of Mathematics. 2020, 41(12), 2320-2325. ISSN 1995-0802.
  • Rok: 2020
  • DOI: 10.1134/S1995080220120148
  • Odkaz: https://doi.org/10.1134/S1995080220120148
  • Pracoviště: Katedra matematiky
  • Anotace:
    In this note we apply noncommutative versions of Lyapunov convexity theorem to obtaning new results in comparison theory of states and functional on von Neumann algebras and JBW* triples. We show that in many cases the sets of projections or tripotents on which functionals attain constant single numerical value are determining for them. We discuss connection of our results with quantum theory.

Finite tripotents and finite JBW*-triples

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Kalenda, Ondrej F. K., Peralta, Antonio M.
  • Publikace: Journal of Mathematical Analysis and Applications. 2020, 490(1), 1-65. ISSN 0022-247X.
  • Rok: 2020
  • DOI: 10.1016/j.jmaa.2020.124217
  • Odkaz: https://doi.org/10.1016/j.jmaa.2020.124217
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study two natural preorders on the set of tripotents in a JB*-triple defined in terms of their Peirce decomposition and weaker than the standard partial order. We further introduce and investigate the notion of finiteness for tripotents in JBW*-triples which is a natural generalization of finiteness for projections in von Neumann algebras. We analyze the preorders in detail using the standard representation of JBW*-triples. We also provide a refined version of this representation - in particular a decomposition of any JBW*-triple into its finite and properly infinite parts. Since a JBW*-algebra is finite if and only if the extreme points of its unit ball are just unitaries, our notion of finiteness differs from the concept of modularity widely used in Jordan structures so far. The exact relationship of these two notions is clarified in the last section. (C) 2020 Elsevier Inc. All rights reserved.

Linearity of Maps on Banach and Operator Algebras

  • DOI: 10.1134/S199508022003018X
  • Odkaz: https://doi.org/10.1134/S199508022003018X
  • Pracoviště: Katedra matematiky
  • Anotace:
    The paper deals with quasi linear maps on two by two matrices over Banach and $$C^{\ast}$$-algebras. Let $$\varphi:A\to X$$ be a homogeneous map between Banach algebra $$A$$ and a linear space $$X$$. Let us take its amplification $$\psi=\varphi^{(2)}$$ to two by two matrix structure $$M_{2}(A)$$ over $$A$$. If $$\psi(x+x^{2})=\psi(x)+\psi(x^{2})$$ for all $$x$$, then $$\varphi$$ is linear. Ramifications for self adjoint parts of Banach $$\ast$$-algebras and $$C^{\ast}$$-algebras as well applications to Mackey–Gleason problem are given.

Measures of weak non-compactness in preduals of von Neumann algebras and JBW*-triples

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Kalenda, Ondrej F. K., Peralta, Antonio M., Pfitzner, H.
  • Publikace: JOURNAL OF FUNCTIONAL ANALYSIS. 2020, 278(1), 1-69. ISSN 0022-1236.
  • Rok: 2020
  • DOI: 10.1016/j.jfa.2019.108300
  • Odkaz: https://doi.org/10.1016/j.jfa.2019.108300
  • Pracoviště: Katedra matematiky
  • Anotace:
    We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW*-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW*-triples with strongly WCG predual and describe the order of seminorms defining the strong* topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW*-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB*-triple into a complex Banach space. (C) 2019 Elsevier Inc. All rights reserved.

The order topology on duals of C*-algebras and von Neumann algebras

  • DOI: 10.4064/sm190108-11-7
  • Odkaz: https://doi.org/10.4064/sm190108-11-7
  • Pracoviště: Katedra matematiky
  • Anotace:
    For a von Neumann algebra M, we study the order topology associated to the hermitian part M-*(s), and to intervals of the predual M-*. It is shown that the order topology on M-*(s) coincides with the topology induced by the norm. In contrast, it is proved that the condition of having the order topology, associated to the interval [0, phi], equal to the topology induced by the norm, for every phi is an element of M-*(+), is necessary and sufficient for the commutativity of M. It is also proved that if phi is a positive bounded functional on a C*-algebra A, then the norm-null sequences in [0, phi] coincide with the null sequences, with respect to the order topology on [0, phi], if and only if the von Neumann algebra pi(phi)(A)' is of finite type (where pi(phi) denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology, on order-bounded parts of dual spaces, are inequivalent for all C*-algebras that are not of type I.

Additivity of Quadratic Maps on JB Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: Lobachevskii Journal of Mathematics. 2019, 40(10), 1483-1488. ISSN 1995-0802.
  • Rok: 2019
  • DOI: 10.1134/S1995080219100123
  • Odkaz: https://doi.org/10.1134/S1995080219100123
  • Pracoviště: Katedra matematiky
  • Anotace:
    The problem of automatic additivity of Jordan type homomorphisms has received a great deal of attention in theory of operator algebras as well in ring theory. We study additivity of quadratic maps, that is the maps between Jordan Banach algebras that preserve the quadratic product (a, b) -> aba. The main result shows that any continuous quadratic bijective map between unital Jordan Banach algebras that is linear on associative subalgebras is automatically additive. This contributes to the Borification program in quantum theory and Mackey-Gleason problem on linearity of quasi-linear maps.

Measures of weak non-compactness in spaces of nuclear operators

  • DOI: 10.1007/s00209-019-02264-2
  • Odkaz: https://doi.org/10.1007/s00209-019-02264-2
  • Pracoviště: Katedra matematiky
  • Anotace:
    We show that in the space of nuclear operators from to (where ) the two natural ways of measuring weak non-compactness coincide. We also provide explicit formulas for these measures. As a consequence the same is proved for preduals of atomic von Neumann algebras.

Choquet Order and Jordan Maps

Piecewise *-homomorphisms and Jordan maps on C*-algebras and factor von Neumann algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Journal of Mathematical Analysis and Applications. 2018, 462(1), 1014-1031. ISSN 0022-247X.
  • Rok: 2018
  • DOI: 10.1016/j.jmaa.2017.12.056
  • Odkaz: https://doi.org/10.1016/j.jmaa.2017.12.056
  • Pracoviště: Katedra matematiky
  • Anotace:
    We investigate maps between C*-algebras that are well behaved with respect to mutually commuting elements. We contribute to the Mackey-Gleason problem by showing that any continuous bijection between self-adjoint parts of C*-algebras that preserves triple product (a, b) -> aba, and is linear on commutative subspaces, is already linear. This allows us to describe such maps as direct differences of linear Jordan isomorphisms. We shall show that any weak*-continuous bijection between positive invertible elements of von Neumann factors (of dimension at least 9) that preserves products of commuting elements in both directions is of the form a -> e(Psi(log a))theta(a(c)), where theta is a linear Jordan *-isomorphism, c nonzero real number and Psi is a hermitian continuous functional. In a similar way we describe the same type of bicontinuous maps between unitary groups of von Neumann factors. General form of the above mentioned maps on C*-algebras is also presented. (C) 2017 Elsevier Inc. All rights reserved.

PREDUALS OF JBW*-TRIPLES ARE 1-PLICHKO SPACES

  • DOI: 10.1093/qmath/hax057
  • Odkaz: https://doi.org/10.1093/qmath/hax057
  • Pracoviště: Katedra matematiky
  • 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.

Structure of abelian parts of C*-algebras and its preservers

  • DOI: 10.14232/actasm-017-582-8
  • Odkaz: https://doi.org/10.14232/actasm-017-582-8
  • Pracoviště: Katedra matematiky
  • Anotace:
    The context poset of Abelian C*-subalgebras of a given C*-algebra is an operator theoretic invariant of growing interest. We review recent results describing order isomorphisms between context posets in terms of Jordan type maps (linear or not) between important types of operator algebras. We discuss the important role of the generalized Gleason theorem on linearity of maps preserving linear combinations of commuting elements for studying symmetries of context posets. Related results on maps multiplicative with respect to commuting elements are investigated.

Decompositions of preduals of JBW and JBW algebras

  • DOI: 10.1016/j.jmaa.2016.08.031
  • Odkaz: https://doi.org/10.1016/j.jmaa.2016.08.031
  • Pracoviště: Katedra matematiky
  • 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.

LINEAR ALGEBRAIC PROOF OF WIGNER THEOREM AND ITS CONSEQUENCES

  • DOI: 10.1515/ms-2016-0273
  • Odkaz: https://doi.org/10.1515/ms-2016-0273
  • Pracoviště: Katedra matematiky
  • Anotace:
    We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan *-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems. (C) 2017 Mathematical Institute Slovak Academy of Sciences

Operational independence and tensor products of C*-algebras

  • DOI: 10.1063/1.4978869
  • Odkaz: https://doi.org/10.1063/1.4978869
  • Pracoviště: Katedra matematiky
  • Anotace:
    Complete C*-independence of operator algebras is introduced. Equivalent characterization is given for C*-subalgebras to be completely independent in terms of maximal tensor product. Besides, the independence of Banach algebras is considered, and we showed that Hahn-Banach independence is a generalization of C*-independence and discussed Hahn-Banach independence in M-n(A), where A is a C*-algebra. Among others, we characterize independence of operator algebras by projective and injective C*-tensor product in terms of simultaneous extensions of completely positive maps. Published by AIP Publishing.

Quantum Spectral Symmetries

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: International Journal of Theoretical Physics. 2017, 56(12), 3807-3818. ISSN 0020-7748.
  • Rok: 2017
  • DOI: 10.1007/s10773-017-3312-z
  • Odkaz: https://doi.org/10.1007/s10773-017-3312-z
  • Pracoviště: Katedra matematiky
  • Anotace:
    Quantum symmetries of spectral lattices are studied. Basic properties of spectral order on AW*-algebras are summarized. Connection between projection and spectral automorphisms is clarified by showing that, under mild conditions, any spectral automorphism is a composition of function calculus and Jordan *-automorphism. Complete description of quantum spectral symmetries on Type I and Type II AW*-factors are completely described.

Bell Correlated and EPR States in the Framework of Jordan Algebras

  • DOI: 10.1007/s10701-015-9966-6
  • Odkaz: https://doi.org/10.1007/s10701-015-9966-6
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study Bell inequalities and EPR states in the context of Jordan algebras. We show that the set of states violating Bell inequalities across two operator commuting nonmodular Jordan Banach algebras is norm dense in the global state space. It generalizes hitherto known results in quantum field theory in several directions. We propose new Jordan quantity for incommensurable observables in a given state, introduce the concept of EPR state for Jordan structures, and study relationship between EPR states and Bell correlated states. Our analysis shows crucial role of spin factors and Pauli spin matrices for studying noncommutative properties of states and observables.

Boundedness of completely additive measures with application to 2-local triple derivations

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Kudaybergenov, K., Peralta, A.M., Russo, B.
  • Publikace: Journal of Mathematical Physics. 2016, 57(2), 1-22. ISSN 0022-2488.
  • Rok: 2016
  • DOI: 10.1063/1.4941988
  • Odkaz: https://doi.org/10.1063/1.4941988
  • Pracoviště: Katedra matematiky
  • Anotace:
    We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measures and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation. 2-local triple derivations are well understood on von Neumann algebras. JBW*-triples, which are properly defined in Section I, are intimately related to infinite dimensional holomorphy and include von Neumann algebras as special cases. In particular, continuous JBW*-triples can be realized as subspaces of continuous von Neumann algebras which are stable for the triple product xy*z + zy*x and closed in the weak operator topology. (C) 2016 AIP Publishing LLC.

COMPLETENESS OF GELFAND-NEUMARK-SEGAL INNER PRODUCT SPACE ON JORDAN ALGEBRAS

  • DOI: 10.1515/ms-2015-0150
  • Odkaz: https://doi.org/10.1515/ms-2015-0150
  • Pracoviště: Katedra matematiky
  • Anotace:
    The paper deals with inner product spaces generated by states on Jordan algebras. We show an interplay between completeness of the Gelfand-Neumark-Segal representation space, geometric properties of states on Jordan algebras, structure of irreducible Jordan representations, and properties of normal states on second duals of Jordan algebras. We prove that if the GNS representation space is complete, then given state must be a convex combination of pure states. On the other hand, we analyze structure of inner product spaces arising from states on spin factors and Type In, n = 4, factors, showing their completeness as a consequence. (C) 2016 Mathematical Institute Slovak Academy of Sciences

On Markushevich bases in preduals of von Neumann algebras

  • DOI: 10.1007/s11856-016-1365-y
  • Odkaz: https://doi.org/10.1007/s11856-016-1365-y
  • Pracoviště: Katedra matematiky
  • 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.

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: International Journal of Theoretical Physics. 2016, 55(7), 3353-3365. ISSN 0020-7748.
  • Rok: 2016
  • DOI: 10.1007/s10773-016-2964-4
  • Odkaz: https://doi.org/10.1007/s10773-016-2964-4
  • Pracoviště: Katedra matematiky
  • Anotace:
    An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita's theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan au-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for sigma-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.

Spectral order on AW*-algebras and its preservers

  • DOI: 10.1134/S1995080216040107
  • Odkaz: https://doi.org/10.1134/S1995080216040107
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study the spectral order on the set of positive contractions in an AW*-algebra. We introduce the concept of lattice theoretic center of the resulting spectral lattice and show that it coincides with the algebraic center of the underlying AW*-algebra A if A is finite. By applying this result we generalize hitherto known characterizations of preserves of the spectral order by showing that any bijection φ acting on the spectral lattice of a finite AW*-algebra that preserves spectral order and orthogonality in both directions is a composition of function calculus and a Jordan *-isomorphism. We show that this result holds in a wide context of all AW*-algebras provided that φ preserves in addition the multiples of unity.

Star order on operator and function algebras and its nonlinear preservers

  • DOI: 10.1080/03081087.2016.1164661
  • Odkaz: https://doi.org/10.1080/03081087.2016.1164661
  • Pracoviště: Katedra matematiky
  • 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

  • Pracoviště: Katedra matematiky
  • 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.

Dye's Theorem and Gleason's Theorem for AW*-algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Journal of Mathematical Analysis and Applications. 2015, 422(2), 1103-1115. ISSN 0022-247X.
  • Rok: 2015
  • DOI: 10.1016/j.jmaa.2014.09.040
  • Odkaz: https://doi.org/10.1016/j.jmaa.2014.09.040
  • Pracoviště: Katedra matematiky
  • Anotace:
    We prove that any map between projection lattices of AW*-algebras A and B, where A has no Type I-2 direct summand, that preserves orthocomplementation and suprema of arbitrary elements, is a restriction of a normal Jordan *-homomorphism between A and B. This allows us to generalize Dye's Theorem from von Neumann algebras to AW*-algebras. We show that Mackey-Gleason-Bunce-Wright Theorem can be extended to homogeneous AW*-algebras of Type I. The interplay between Dye's Theorem and Gleason's Theorem is shown. As an application we prove that Jordan *-homomorphisms are commutatively determined. Another corollary says that Jordan parts of AW*-algebras can be reconstructed from posets of their abelian subalgebras.

The order topology for a von Neumann algebra

  • DOI: 10.4064/sm8041-1-2016
  • Odkaz: https://doi.org/10.4064/sm8041-1-2016
  • Pracoviště: Katedra matematiky
  • Anotace:
    The order topology tau(o)(P) (resp. the sequential order topology tau(os)(P)) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part M-sa, the self-adjoint part of the unit ball M-sa(1), and the projection lattice P(M). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on M, and relate the properties of the order topology to the underlying operator-algebraic structure of M.

Automorphisms of Ordered Structures of Abelian Parts of Operator Algebras and their Role in Quantum Theory

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: International Journal of Theoretical Physics. 2014, 53(10), 3333-3345. ISSN 0020-7748.
  • Rok: 2014
  • DOI: 10.1007/s10773-013-1691-3
  • Odkaz: https://doi.org/10.1007/s10773-013-1691-3
  • Pracoviště: Katedra matematiky
  • Anotace:
    It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.

Classes of Invariant Subspaces for Some Operator Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: International Journal of Theoretical Physics. 2014, 53(10), 3397-3408. ISSN 0020-7748.
  • Rok: 2014
  • DOI: 10.1007/s10773-013-1740-y
  • Odkaz: https://doi.org/10.1007/s10773-013-1740-y
  • Pracoviště: Katedra matematiky
  • Anotace:
    New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace L such that all important subspace classes living on L are different. Moreover, we show that L can be chosen such that the set of σ-additive measures on subspace classes of L are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.

Comment on: "Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics" [Phys. Lett. A 372 (2008) 6847]

  • Autoři: Simon, R., Mukunda, N., Chaturvedi, S., Srinivasan, V., prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Physics Letters A. 2014, 30-31(378), 2332-2335. ISSN 0375-9601.
  • Rok: 2014
  • DOI: 10.1016/j.physleta.2014.03.058
  • Odkaz: https://doi.org/10.1016/j.physleta.2014.03.058
  • Pracoviště: Katedra matematiky
  • Anotace:
    The second of the two proofs of the Wigner theorem on symmetry in quantum mechanics given in Simon et al. [Phys. Lett. A 372 (2008) 6847] is reexamined and refined to remove a possible gap in the argument presented there.

Star order on JBW algebras

  • DOI: 10.1016/j.jmaa.2014.03.054
  • Odkaz: https://doi.org/10.1016/j.jmaa.2014.03.054
  • Pracoviště: Katedra matematiky
  • 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.

Affilated subspaces and infiniteness of von Neumann algebras

  • DOI: 10.1002/mana.201200157
  • Odkaz: https://doi.org/10.1002/mana.201200157
  • Pracoviště: Katedra matematiky
  • Anotace:
    We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi-splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi-splitting subspaces are non-equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi-splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.

Affiliated subspaces and the structure of von neumann algebras

  • DOI: 10.7900/jot.2010jul07.1894
  • Odkaz: https://doi.org/10.7900/jot.2010jul07.1894
  • Pracoviště: Katedra matematiky
  • Anotace:
    The interplay between order-theoretic properties of structures of subspaces affiliated with a von Neumann algebra M and the inner structure of the algebra M is studied. The following characterization of finiteness is given: a von Neumann algebra M is finite if and only if in each representation space of M one has that closed affiliated subspaces are given precisely by strongly closed left ideals in M. Moreover, it is shown that if the modular operator of a faithful normal state φ is bounded, then all important classes of affiliated subspaces in the GNS representation space of φ coincide. Orthogonally closed affiliated subspaces are characterized in terms of the supports of normal func-tionals. It is proved that complete affiliated subspaces correspond to left ideals generated by finite sums of orthogonal atomic projections

Nonlinear maps on von Neumann algebras preserving the star order

  • DOI: 10.1080/03081087.2012.721363
  • Odkaz: https://doi.org/10.1080/03081087.2012.721363
  • Pracoviště: Katedra matematiky
  • 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

Structure of associative subalgebras of Jordan operator algebras

  • DOI: 10.1093/qmath/has015
  • Odkaz: https://doi.org/10.1093/qmath/has015
  • Pracoviště: Katedra matematiky
  • Anotace:
    We show that any order isomorphism between ordered structures of associative unital JB-subalgebras of JBW algebras is implemented naturally by a Jordan isomorphism. Consequently, JBW algebras are determined by the structure of their associative unital JB subalgebras. Further we show that in a similar way it is possible to reconstruct Jordan structure from the order structure of associative subalgebras endowed with an orthogonality relation. In case of abelian subalgebras of von Neuman algebra it is we shown that order isomorphisms of the structure of abelian C*-subalgebras that are well behaved with respect to the structure of two by two matrices over original algebra are implemented by *-isomorphisms.

COMPLETENESS OF *-SYMMETRIC GELFAND-NAIMARK-SEGAL INNER PRODUCT SPACES

  • DOI: 10.1093/qmath/haq041
  • Odkaz: https://doi.org/10.1093/qmath/haq041
  • Pracoviště: Katedra matematiky
  • Anotace:
    Every state rho on a C*-algebra A induces a *-symmetric semi-inner product (x, y)& rho(y* x) + rho(xy*) (x, y is an element of A). The main scope of the paper is to characterize those states for which the induced *-symmetric Gelfand-Naimark-Segal inner product space is complete. It is shown that this happens precisely when rho is a finite convex combination of pure states. (It is well known that the same conclusion follows if one considers the non-symmetric semi-inner product (x, y) & rho(y* x).) In so doing, we exhibit an interesting connection between convexity properties of states, the transitivity of irreducible representations and Banach space properties of the quotients of C*-algebra s by kernels of states.

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

  • DOI: 10.1063/1.4771671
  • Odkaz: https://doi.org/10.1063/1.4771671
  • Pracoviště: Katedra matematiky
  • Anotace:
    In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only one numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molná

Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Journal of Mathematical Analysis and Its Applications. 2011, 383(2), 391-399. ISSN 0022-247X.
  • Rok: 2011
  • DOI: 10.1016/j.jmaa.2011.05.035
  • Odkaz: https://doi.org/10.1016/j.jmaa.2011.05.035
  • Pracoviště: Katedra matematiky
  • Anotace:
    The aim of this note is to study the interplay between the Jordan structure of C*-algebra and the structure of its abelian C*-subalgebras. Let Abel(A) be a system of unital C*-subalgebras of a unital C*-algebra A ordered by set theoretic inclusion. We show that any order isomorphism phi : Abel(A) -> Abel(B) can be uniquely written in the form phi(C) = psi(C(sa)) + i psi (C(sa)), where psi is a partially linear Jordan isomorphism between self-adjoint parts of unital C*-algebras A and B. As a corollary we obtain that for certain class of C*-algebras (including von Neumann algebras) ordered structure of abelian subalgebras completely determines the Jordan structure. The results extend hitherto known results for abelian C*-algebras and may be relevant to foundations of quantum theory. (C) 2011 Elsevier Inc. All rights reserved.

Operator Algebras and Quantum Structures

  • Pracoviště: Katedra matematiky
  • 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.

Subspace Structures in Inner Product Spaces and von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Turilova, E.
  • Publikace: International Journal of Theoretical Physics. 2011, 2011(50), 3812-3820. ISSN 0020-7748.
  • Rok: 2011
  • DOI: 10.1007/s10733-011-0665-6
  • Odkaz: https://doi.org/10.1007/s10733-011-0665-6
  • Pracoviště: Katedra matematiky
  • Anotace:
    We study subspaces of inner product spaces that are invariant with respect to a given von Neumann algebra. The interplay between order properties of the poset of affiliated subspaces and the structure of a von Neumann algebra is investigated. We extend results on nonexistence of measures on incomplete structures to invariant subspaces. Results on inner product spaces as well as on the structure of affiliated subspaces are reviewed.

Absolute Continuity and Noncommutative Measure Theory

  • DOI: 10.1007/s10773-009-0105-z
  • Odkaz: https://doi.org/10.1007/s10773-009-0105-z
  • Pracoviště: Katedra matematiky
  • Anotace:
    Recent results on absolute continuity of Banach space valued operators and convergence theorems on operator algebras are deepened and summarized It is shown that absolute continuity of an operator T on a von Neumann algebra M with respect to a positive normal functional psi on M is not implied by the fact that the null projections of psi are the null projections of T However, it is proved that the implication above is true whenever M is finite or T is weak*-continuous Further it is shown that the absolute value preserves the Vitali-Hahn-Saks property if, and only if, the underlying algebra is finite This result Improves classical results on weak compactness of sets of noncommutative measures

Bell's Correlations and Spin Systems

  • DOI: 10.1007/s10701-009-9401-y
  • Odkaz: https://doi.org/10.1007/s10701-009-9401-y
  • Pracoviště: Katedra matematiky
  • 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.

Independence of Group Algebras

  • DOI: 10.1002/mana.200710038
  • Odkaz: https://doi.org/10.1002/mana.200710038
  • Pracoviště: Katedra matematiky
  • Anotace:
    It is shown that major independence conditions for left and right group operator algebras coincide. If Gamma is a discrete ICC group, then the reduced left and right group algebras W-lambda*(F) and W-phi*(Gamma) are W*-independent. These algebras are moreover independent in the product sense if, and only if, r is amenable. If A and B are subgroups of Gamma, then the left and right reduced group (sub)algebrasW(lambda)*(A) and W-phi*(B) are W*-independent provided that any of the following two conditions is satisfied: (i) A and B have trivial intersection; (ii) A or B is ICC. The results indicate an interplay between intrinsic group-theoretic properties and independence of the corresponding group algebras that can be further exploited. New examples of W*-independent von Neumann algebras arising from groups are generated.

A noncommutative Brook-Jewett Theorem

  • Autoři: Chetcutti, E., prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Journal of Mathematical Analysis and Its Applications. 2009, 2009(355), 839-845. ISSN 0022-247X.
  • Rok: 2009
  • DOI: 10.1016/j.jmaa.2009.02.018
  • Odkaz: https://doi.org/10.1016/j.jmaa.2009.02.018
  • Pracoviště: Katedra matematiky
  • Anotace:
    In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks- Jewett property if, and only if, it is topologically equivalent to an abelian algebra.

Bell's Inequalities and Pauli Matrices

  • Pracoviště: Katedra matematiky
  • Anotace:
    Bell's inequalities and their maximal violators are investigated.

Generalization of Bell's inequalities

  • Pracoviště: Katedra matematiky
  • Anotace:
    Bell's inequalities are generalized to *-algebras. The interesting structural consequences are investigated.

Maximal violation of Bell's inequalities and Pauli spin matrices

  • DOI: 10.1063/1.3190118
  • Odkaz: https://doi.org/10.1063/1.3190118
  • Pracoviště: Katedra matematiky
  • 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.

NON-COMMUTATIVE VITALI-HAHN-SAKS THEOREM HOLDS PRECISELY FOR FINITE W*-ALGEBRAS

Spectral Lattices

Quantum Structures and Operator Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Handbook of Quantum Logic and Quantum Structures. Amsterdam: Elsevier, 2007. p. 285-333. ISBN 978-0-444-52870-4.
  • Rok: 2007

Spectral Order of Operators and Range Projections

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Journal of Mathematical Analysis and Its Applications. 2007, 331(2), 1122-1134. ISSN 0022-247X.
  • Rok: 2007

Vitali-Hahn-Saks-Theorem For Vector Measures on Operator Algebras

  • Autoři: Chetcuti, E., prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: The Quarterly Journal of Mathematics. 2006, 2006(57), 479-493. ISSN 0033-5606.
  • Rok: 2006

Orthogonal Pure States in Operator Theory

  • Pracoviště: Katedra matematiky
  • Anotace:
    New generalizations of basic topological principles of theory of locally compact spaces to JB algebras and von Neumann algebras are found. The results are applied to quatum theory.

States on Operator Algebras and Axiomatic System of Quantum Theory

  • Pracoviště: Katedra matematiky
  • Anotace:
    New results on the hidden variables for von Neumann algebras are presented. The structure of subsystems in general theory of operator algebras as well as quantum field theory is investigated.

C*-Independence, Product States and Commutation

  • Pracoviště: Katedra matematiky
  • Anotace:
    The interplay between commutation phenomena, product type extensions and C*-independence of C*-algebras is studied. New characterization of tensor product of C*-algebras in terms of extension properties is given. Results are applied to quantum field theory

States and Structure of von Neumann Algebras

  • Pracoviště: Katedra matematiky
  • Anotace:
    New results on the states on projection structure of von Neumann algebra and their connection to structure theory of are deepened and summarized. The first part deals with Jauch-Piron states. The second part is devoted to independence of operator algebras

States on Operator Algebras and Axiomatics of Quantum Theory

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Proceedings of the Quantum Structures 2004. New Mexico: New Mexico State University, 2004. pp. 6.
  • Rok: 2004
  • Pracoviště: Katedra matematiky
  • Anotace:
    New results on the theory of states on operator algebras and their application to mathematical foundations of quantum theory are presented

Traces, Dispersions and Hidden Variables

  • Pracoviště: Katedra matematiky
  • Anotace:
    It is shown that dispersion of states on C*-algebras with rich projection structure is universally bounded from zero. Traces on von Neumann algebras are characterized in terms of dispersions. The results are applied to theory of hidden variables in quantum mechanics

Quantum Measure Theory

  • Pracoviště: Katedra matematiky
  • Anotace:
    The book is the first systematic treatment of measures on projection lattices of von Neuman algebras. It presents significant recent results in the theory of operator algebras and mathematical foundations of quantum theory.

C*-independence and W*-independence of von Neumann Algebras

De Morgan property for effect algebras of von Neumann algebras

Multiplicativity of Extremal Positive Maps on Abelian Parts of Operator Algebras

Noncommutative Phenomena in Measure Theory on Operator Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Rendiconti dell'Istituto di Matematica dell'Universita di Trieste. 2002, 2002(34), 19-43. ISSN 0049-4704.
  • Rok: 2002

Independence in Quantum Probability Theory

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Proceedings of Summer School on Real Analysis and Measure Theory. Trieste: Universitá Trieste, 2001. pp. 21-25.
  • Rok: 2001

Pure States on Jordan Algebras

States and Structure of von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Quantum Structures V. Cesena-Cesenatico: International Quantum Structures Association, 2001. pp. 42-43.
  • Rok: 2001

Jauch-Piron States and Sigma-Additivity

On the de Morgan Property of the Standard Brouwer-Zadeh Poset

Restricting and Extending States and Positive Maps on Operator Algebras

Measure and Integration Theory on Operator Algebras-non-commutative Phenomena I, II

Operator Algebras and Ordered Sets with Orthocomplementations

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Proceedings of Workshop 99. Praha: České vysoké učení technické v Praze, 1999. pp. 42.
  • Rok: 1999

Supporting Sequences of Pure States on JB Algebras

Universal State Space Embeddability of Jordan-Banach Algebras

Determinacy of States and Independence of Operator Algebras

  • DOI: 10.1023/A:1026697003005
  • Odkaz: https://doi.org/10.1023/A:1026697003005
  • Pracoviště: Katedra matematiky
  • Anotace:
    The aim of this paper is to summarize, deepen, and comment upon recent results concerning states on operator algebras and their extensions. The first part is focused on the relationship between pure states and singly generated subalgebras. Among others we show that every pure state rho on a separable algebra A is uniquely determined by some element of A which exposes rho. The main part of this paper is the second section, dealing with characterization of various types of independence conditions arising in the axiomatics of quantum field theory. These two topics, seemingly different, are connected by a common extension technique based on determinacy of pure states.

Restricting Pure States on JB Algebras to Maximal Associative Subalgebras

Statistical Independence of Operator Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Annales de l'Institut Henri Poincaré, Physique théorique. 1997, 67(4), 447-462. ISSN 0246-0211.
  • Rok: 1997
  • Pracoviště: Katedra matematiky
  • Anotace:
    In the paper we investigate statistical independence of C*-algebras and its relation to other independence conditions studied in operator algebras and quantum field theory. Especially, we prove that C*-algebras A(1) and A(2) are statistically independent if and only if for every normalized elements a is an element of A(1) and b is an element of A(2) there is a state phi of the whole algebra such that phi(a) = phi(b) = 1. As a consequence we show that logical independence (see [17, 18]) implies statistical independence and that statistical independence implies independence in the sense of Schlieder. We prove that the reverse implications are not valid, Further, independence of commuting algebras is shown to be equivalent to independence of their centers. Finally, results on independence of commuting algebras are generalized to the context of Jordan-Banach algebras.

The Nikodym Baundedness Theorem

Extensions of Jauch-Piron States on Jordan Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Bunce, L.
  • Publikace: Mathematical Proceedings of the Cambridge Philosophical Society. 1996, 119(2), 279-286. ISSN 0305-0041.
  • Rok: 1996

Extensions Theorems (Vector Measures on Quantum Logic)

Quantum Field Theory and Extensions of States

Countably Additive Homomorphisms Between von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Bunce, L.
  • Publikace: Proceedings of the American Mathematical Society. 1995, 123(11), 3437-3441. ISSN 0002-9939.
  • Rok: 1995

Extension Properties of States on Operator Algebras

Extensions of States on Operator Algebras

States on orthoalgebras

Traces and Subadditive Measures on Projections in JBW-algebras and von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc., Bunce, L.
  • Publikace: Proceedings of the American Mathematical Society. 1995, 123(1), 157-160. ISSN 0002-9939.
  • Rok: 1995
  • DOI: 10.2307/2160621
  • Odkaz: https://doi.org/10.2307/2160621
  • Pracoviště: Katedra matematiky
  • Anotace:
    Let P(M) be the projection lattice of an arbitrary JEW-algebra or von Neumann algebra M. It is shown that the tracial states of M correspond by extension precisely to the subadditive probability measures on P(M). The analogous result for normal semifinite traces is also proved.

Gleason Property and Extensions of States on Projection Logics

Jauch-Piron States on von Neumann Algebras

States on Von Neumann Algebras and Non-commutative Measure Theory

Measures on von Neumann Algebras

Pure Jauch-Piron States on von Neumann Algebras

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Annales de l'Institut Henri Poincaré, Physique théorique. 1993, 58(2), 173-187. ISSN 0246-0211.
  • Rok: 1993

States on Projection Logics of Operator Algebras

STATES ON PROJECTION LOGICS OF VON-NEUMANN-ALGEBRAS

  • Pracoviště: Katedra matematiky
  • Anotace:
    We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.

Von Neumann Algebras and Noncommutative Measure Theory

Additivity of Vector Gleason Measures

Centrally Determined States on Von Neumann Algebras

Completeness and Modular Cross-symmetry in Normed Linear Spaces

Hilbert-space-valued States on Guantum Logics

Hilbert-space-valued Measures on Boolean Algebras (Extensions)

  • Pracoviště: Katedra matematiky
  • Anotace:
    We prove that if B1 is a Boolean subalgebra of B2 and if m: B1 ! H is a bounded finitely additive measure, where H is a Hilbert space, then m admits an extension over B2. This result generalizes the well-known result for real-valued measures (see e.g. [1]). Then we consider orthogonal measures as a generalization of two-valued measures. We show that the latter result remains valid for dimH < 1. If dimH = 1, we are only able to prove a weaker result: If B1 is a Boolean subalgebra of B2 and m: B1 ! H is an orthogonal measure, then we can find a Hilbert space K such that H K and such that there is an orthogonal measure k : B2 ! K with k/B1 = m.

Orthosymmetry and Modularity in Ortholattices

A Representation of Finitely-modular AC-lattices

Orthogonal Vector Measures on Projection Lattices in a Hilbert Space

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Commentationes Mathematicae Universitatis Carolinae. 1990, 1990(31), 655-660. ISSN 0010-2628.
  • Rok: 1990

States on W*-algebras and Orthogonal Vector Measures

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Proceedings of the American Mathematical Society. 1990, 1990(110), 803-806. ISSN 0002-9939.
  • Rok: 1990

On Modular Spaces

  • Autoři: prof. RNDr. Jan Hamhalter, CSc.,
  • Publikace: Bulletin of the Polish Academy of Sciences. Mathematics. 1989, 1989(37), 647-653. ISSN 0239-7269.
  • Rok: 1989

The Sums of Closed Subspaces in a Topological Linear Space

A completeness Criterion for Inner Product-Spaces

On Uniformly Rotund Spaces

Za stránku zodpovídá: Ing. Mgr. Radovan Suk