Emmanuel Chetcuti

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.

A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.

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.

The Existence of Finitely Additive States on Orthogonally Closed Subspaces of Incomplete Inner Product Spaces

We give a positive answer to an old problem of whether there exists an incomplete inner product space S such that its system of orthogonally closed subspaces—denoted by F(S)—admits finitely additive states. Indeed, we show that every infinite-dimensional separable Hilbert space H contains an incomplete dense hyperplane S ⊂ H such that F(S) admits finitely additive states. We also show that the system of orthogonally closed subspaces of any inner product space with countably infinite linear dimension always admits finitely additive states.

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...

Measures on the Splitting Subspaces of an Inner Product Space

Let S be an inner product space and let E(S) (resp. F(S)) be the orthocomplemented poset of all splitting (resp. orthogonally closed) subspaces of S. In this article we study the possible states/charges that E(S) can admit. We first prove that when S is an incomplete inner product space such that dim S/S < ∞, then E(S) admits at least one state with a finite range. This is very much in contrast to states on F(S). We then go on showing that two-valued states can exist on E(S) not only in the case when E(S) consists of the complete/cocomplete subspaces of S. Finally we show that the well known result which states that every regular state on L(H) is necessarily σ-additive cannot be directly generalized for charges and we conclude by giving a sufficient condition for a regular charge on L(H) to be σ-additive.

Range of Charges on Orthogonally Closed Subspaces of an Inner Product Space

It is still an open question whether the complete lattice F(S) of all orthogonally closed subspaces of an incomplete inner product space S admits a nonzero charge. A negative answer would result in a new way of completeness characterization of inner product spaces. Many partial results have been established regarding what has now turned to be a highly nontrivial problem. Recently, in Dvurečenskij and Ptak (Letters in Mathematical Physics, 62, 63–70, 2002) the range of a finitely additive state s on F(S), dim S = ∞, was shown to include the whole interval [0, 1]. This was then generalized in Dvurečenskij (International Journal of Theoretical Physics, 2003) for general inner product spaces satisfying the Gleason property. Motivated by these results, we give a thorough investigation of the possible ranges of charges on F(S), dim S ≥q3. We show that if the nonzero charge m is bounded, then for infinite dimensional inner product spaces, Range(m) is always convex. We also show that this need not be the case with unbounded charges. Finally, in the last section, we investigate the range of charges on F(S), dim S = ∞, satisfying the sign-preserving and Jauch-Piron properties. We show that for such measures the range is again always convex.

Boundedness of sign-preserving charges, regularity, and the completeness of inner product spaces

We introduce sign-preserving charges on the system of all orthogonally closed subspaces, F ( S ), of an inner product space S , and we show that it is always bounded on all the finite-dimensional subspaces whenever dim S = ∞. When S is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that S is complete whenever F ( S ) admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.

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.