David Buhagiar

Loomis--Sikorski representation of monotone σ-complete effect algebras

We show that monotone @s-complete effect algebras with the Riesz decomposition property are @s-homomorphic images of effect-tribes with the Riesz decomposition property, which are effect algebras of fuzzy sets closed under pointwise limits of nondecreasing fuzzy sets.

Algebraic and measure-theoretic properties of classes of subspaces of an inner product space

Publisher Summary In the classical Kolmogorov probability theory, the set of experimentally verifiable events assigned to physical systems can be identified with a measurable space. The order relation ≤ induced by the lattice operations V and Λ is logically interpreted as the implication relation. A probability measure is a countably additive, normalized and nonnegative function μ on Σ. Random variables are the Σ-measurable real-valued functions on X. The algebraic study of quantum logics that generalize Boolean σ-algebras has given rise to the theory of orthomodular posets, and the study of states to non-commutative measure theory. One of the most important quantum logic is the projection lattice of a Hilbert space H . The basic axiom of the Hilbert space model is that the events of a quantum system can be represented by projections on a Hilbert space or, equivalently, the collection L (H) of closed subspaces of a Hilbert space H . The transition from the Boolean σ-algebra Σ to the projection lattice L (H) consists in replacing the disjointedness of sets by a geometric concept of the orthogonality of subspaces.

ONLY 'FREE' MEASURES ARE ADMISSABLE ON F(S) WHEN THE INNER PRODUCT SPACE S IS INCOMPLETE

Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space S is a sufficient (and of course, a necessary) condition for S to be a Hilbert space.

On metrizable type (MT-) maps and spaces

In this paper we define and study MT-maps, which are the fibrewise topological analogue of metrizable spaces, i.e., the extension of metrizability from the category Top to the category Top Y . Several characterizations and properties of MT-maps are proved. The notion of an MT-space as an MT-map preimage of a metrizable space is introduced. Examples of MT-spaces and their relation withM-spaces are given. Finally it is deduced that an MT-space with a G -diagonal is metrizable.

Quasi-splitting subspaces and Foulis-Randall subspaces

For a pre-Hilbert space S, let F(S) denote the orthogonally closed subspaces, Eq(S) the quasi-splitting subspaces, E(S) the splitting subspaces, D(S) the Foulis-Randall subspaces, and R(S) the maximal Foulis-Randall subspaces, of S. It was an open problem whether the equalities D(S) = F(S) and E(S) = R(S) hold in general [Cattaneo, G. and Marino, G., “Spectral decomposition of pre-Hilbert spaces as regard to suitable classes of normal closed operators,” Boll. Unione Mat. Ital. 6 1-B, 451–466 (1982); Cattaneo, G., Franco, G., and Marino, G., “Ordering of families of subspaces of pre-Hilbert spaces and Dacey pre-Hilbert spaces,” Boll. Unione Mat. Ital. 71-B, 167–183 (1987); Dvurecenskij, A., Gleasons Theorem and Its Applications (Kluwer, Dordrecht, 1992), p. 243.]. We prove that the first equality is true and exhibit a pre-Hilbert space S for which the second equality fails. In addition, we characterize complete pre-Hilbert spaces as follows: S is a Hilbert space if, and only if, S has an orthonormal basis...

Selections and countable compactness

The present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

κ-compactness, extent and the Lindelöf number in LOTS

We study the behaviour of ℵ-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.

Completeness of Inner Product Spaces and G. Cattaneo

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.

Quasi-uniform completions of partially ordered spaces

AbstractIn this paper we define partially ordered quasi-uniform spaces (X,