Publikace

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.

We show that a (non-negative) measure on a circle coarse-grained system of sets can be extended, as a (non-negative) measure, over the collection of all subsets of the circle. This result contributes to quantum logic probability (de Lucia in Colloq Math 80(1):147-154, 1999; Gudder in Quantum Probability, Academic Press, San Diego, 1988; Gudder in SIAM Rev 26(1):71-89, 1984; Harding in Int J Theor Phys 43(10):2149-2168, 2004; Navara and Ptak in J Pure Appl Algebra 60:105-111, 1989; Ptak in Proc Am Math Soc 126(7):2039-2046, 1998, etc.) and completes the analysis of coarse-grained measures carried on in De Simone and Ptak (Bull Pol Acad Sci Math 54(1):1-11, 2006; Czechoslov Math J 57(132) n.2:737-746, 2007), Gudder and Marchand (Bull Pol Acad Sci Math 28(11-12):557-564, 1980) and Ovchinnikov (Construct Theory Funct Funct Anal 8:95-98, 1992).

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

In this note we continue the investigation of algebraic properties of orthocomplemented (symmetric) difference lattices (ODLs) as initiated and previously studied by the authors. We take up a few identities that we came across in the previous considerations. We first see that some of them characterize, in a somewhat non-trivial manner, the ODLs that are Boolean. In the second part we select an identity peculiar for set-representable ODLs. This identity allows us to present another construction of an ODL that is not set-representable. We then give the construction a more general form and consider algebraic properties of the 'orthomodular support'.

Connections between various generalizations of orthocomplete and lattice effect algebras are shown. Archimedean weakly orthocomplete effect algebras are characterized.

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.

Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3533918]

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.

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.

The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e.g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is MO(3) x 2(4).