A noncommutative Brooks–Jewett Theorem

AbstractIn 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

Quantum Spectral Symmetries

© 2017 Springer Science+Business Media New YorkQuantum 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

Affiliated subspaces and infiniteness of von Neumann algebras

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. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Affiliated subspaces and the structure of von neumann algebras

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. © Theta, 2013

Structure of associative subalgebras of Jordan operator algebras

We show that any order isomorphism between ordered structures of associative unital JB-subalgebras (Jordan-Banach) of JBW algebras (dual Jodan-Banach) is implemented naturally by a Jordan isomorphism. Consequently, JBW algebras are determined by the structure of their associative unital JB subalgebras. Furthermore, 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 Neumann algebras, we show 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. © 2012. Published by Oxford University Press. All rights reserved

Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

© 2013, Springer Science+Business Media New York. 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

© 2013, Springer Science+Business Media New York. 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 ${\mathcal{L}}$ such that all important subspace classes living on ${\mathcal{L}}$ are different. Moreover, we show that ${\mathcal{L}}$ can be chosen such that the set of σ-additive measures on subspace classes of ${\mathcal{L}}$ 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

Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras

© 2016 Mathematical Institute Slovak Academy of Sciences.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