Jiří Janda

On Bilinear Forms from the Point of View of Generalized Effect Algebras

We study positive bilinear forms on a Hilbert space which are not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete generalized effect algebras.

Maximal Subsets of Pairwise Summable Elements in Generalized Effect Algebras

We show that in any generalized effect algebra (G;⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G;⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by its summability blocks, which are generalized MV-effect algebras in the case that G is lattice ordered. We also present the relations between summability blocks and compatibility blocks of G. Counterexamples, to obtain the required contradictions in some cases, are given.

Regular Gleason Measures and Generalized Effect Algebras

We study measures, finitely additive measures, regular measures, and σ-additive measures that can attain even infinite values on the quantum logic of a Hilbert space. We show when particular classes of non-negative measures can be studied in the frame of generalized effect algebras.

Intervals in generalized effect algebras

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Riečanová and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.

Blocks in Pairwise Summable Generalized Effect Algebras

We study the so-called pairwise summable generalized effect algebras and show some conditions under which they are generalized MV-effect algebras. Moreover, we give a necessary and sufficient condition for finite antichains of elements in pairwise summable generalized effect algebras under which they are sets of atoms of sub-generalized effect algebras, which are generalized MV-effect algebras in their own right. Simultaneously, we give counterexamples which show properties of those antichains which are not necessary or sufficient to be atoms of such generalized MV-effect algebras.

Weakly ordered partial commutative group of self-adjoint linear operators densely defined on Hilbert space

ABSTRACT We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space ℋ. In [Paseka, J.- -Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

More on PT-Symmetry in (Generalized) Effect Algebras and Partial Groups

We continue in the direction of our paper on PT -Symmetry in (Generalized) Effect Algebras and Partial Groups. Namely we extend our considerations to the setting of weakly ordered partial groups. In this setting, any operator weakly ordered partial group is a pasting of its partially ordered commutative subgroups of linear operators with a fixed dense domain over bounded operators. Moreover, applications of our approach for generalized effect algebras are mentioned.