Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

Test spaces and characterizations of quadratic spaces

We show that a test space consisting of nonzero vectors of a quadratic spaceE and of the set all maximal orthogonal systems inE is algebraic iffE is Dacey or, equivalently, iffE is orthomodular. In addition, we present another orthomodularity criteria of quadratic spaces, and using the result of Solèr, we show that they can imply thatE is a real, complex, or quaternionic Hilbert space.     

Local Automorphisms of Some Quantum Mechanical Structures

Let H be a separable infinite-dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.     

Field representations of the conformal group with continuous mass spectrum

The discrete series of the conformal groupSU(2, 2) is realized on a Hilbert space of holomorphic functions over a bounded domain or the field theoretic tube domain. The boundary values of these functions form Hilbert spaces of distributions. For the realization over the tube domain the boundary distributions transform like classical spinorial fields with a continuous mass spectrum extending from zero to infinity. The reduction of these field realizations of the whole discrete series into unitary irreducible representations of the inhomogeneous Lorentz group is explicitly given.     

Orthomodular posets of idempotents in finite rings of matrices

The idempotents, resp. Hermitian idempotents, of a unital ring, resp. involutive unital ring, form an orthomodular poset. We study these Orthomodular posets for rings of matrices over the integers modulom or over Galois fields. In analogy to the Hilbert space situation we look for idempotent matrices (projections) corresponding to splitting subspaces of finite-dimensional vector spaces.     

Deformation Quantization for Hilbert Space Actions

Rieffel's theory of deformations of C^*-algebras for -actions can be extended to actions of infinite-dimensional Hilbert spaces. The CCR algebra over a Hilbert space H can be exhibited in this manner as a deformation of a commutative C^*-algebra of almost periodic functions on H. Received: 26 August 1996 / Accepted: 28 January 1997     

Hilbert lattices: New results and unsolved problems

The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on form-topological orthomodular spaces and results on arbitrary orthomodular spaces are appended.Deceased (October 29, 1989).     

Sharp Elements in Effect Algebras

We show that if for an arbitrary pair of orthogonal sharp elements of an effect algebra E its join exists and is sharp, then the set E_S of all sharp elements of E is a subeffect algebra of E that is an orthomodular poset. Such effect algebras need not be sharply dominating but S-dominating. Further, we show that in every nonproper effect algebra E, E_S is a subeffect algebra that is an orthomodular poset. Moreover, a general theorem for E_S is proved.     

Automorphisms of An Orthomodular Poset of Projections

By using a lattice characterization of continuous projections defined on a topological vector space E arising from a dual pair, we determine the automorphism group of their orthomodular poset Proj(E) by means of automorphisms and anti-automorphisms of the lattice L of all closed subspaces of E. A connection between the automorphism group of the ring of all continuous linear mappings defined on E and the automorphism group of the orthoposet Proj(E) is established.     

On the Geometry of Orthomodular Spaces over Fields of Power Series

Orthomodular spaces are generalizations ofHilbert spaces with which they share the basic propertyexpressed by the Projection Theorem. We study twoinfinite-dimensional orthomodular spaces, bothconstructed over the same field of power series, but withdifferent inner products. On the first space everybounded, self-adjoint operator decomposes into anorthogonal sum of operators of rank 1 or 2; on thesecond space, in contrast, there exist self-adjointoperators that are undecomposable. These differencesreflect the fact that the underlying geometries aredissimiliar.     

Properties of Quasi-Hermitian Operators Inherited from Self-Adjoint Operators

We study a generalized effect algebra of unbounded linear operators in an infinite-dimensional complex Hilbert space. This algebra equipped with a certain kind of topology allows us to show that unbounded quasi-Hermitian operators can be expressed as a difference of two infinite sums of bounded quasi-Hermitian operators.     

Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments fQ _f from compactly supported C functions f to bounded linear operators Q _f on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.     

Orthomodularity of Decompositions in a Categorical Setting

We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of familiar mathematical structures such as non-empty sets, groups, and topological spaces, as well as a method to construct an orthomodular poset from the complementary pairs of elements of a bounded modular lattice. Mathematics Subject Classifications (2000): 06C15, 81P10, 03G12, 18A30     

Current algebra representation for the 3+1 dimensional Dirac-Yang-Mills theory

The structure of the current algebra representation in the state space of fermions in an external Yang-Mills field in 3+1 space-time dimensions is analyzed; the topology of the vector space is determined by a countable family of semi-definite inner products. We show that there is no hermitian non-trivial Hilbert space representation such that the energy is bounded from below. The structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.     

Covariance and Quantum Logic

Considering the fundamental role symmetry plays throughout physics, it is remarkable how little attention has been paid to it in the quantum-logical literature. In this paper, we discuss G-test spaces—that is, test spaces hosting an action by a group G—and their logics. The focus is on G-test spaces having strong homogeneity properties. After establishing some general results and exhibiting various specimens (some of them exotic), we show that a sufficiently symmetric G-test space having an invariant, separating set of states with affine dimension n, is always representable in terms of a real Hilbert space of dimension n+1, in such a way that orthogonal outcomes are represented by orthogonal unit vectors.     

Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a minimal support condition (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice (A, ) for a realistic entity (A, ) in the sense of Foulis-Piron-Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.     

Orthomodular Lattices of Subspaces Obtained from Quadratic Forms

Being given a field K of characteristic different from 2 and 3, a 3-dimensional vector space E over K, and a nonsingular symmetric bilinear form φ over E, we define a structure of orthomodular lattice T(E,φ) on the set of all nonisotropic subspaces of E. We give a structure Theorem about the irreducible and 3-homogeneous subalgebras of T(E,φ). In particular, these subalgebras are all of the form T(E',φ ') where E' is a 3-dimensional subspace of E, if E is regarded as a vector space over a subfield K' of K, and φ ' is induced by φ. This structure Theorem allows us to achieve an old project, concerning minimal orthomodular lattices (an orthomodular lattice L is called minimal if it is nonmodular and if all its proper subalgebras are either modular, or isomorphic to L).     

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

Generalized Klauder–Perelomov and Gazeau–Klauder coherent states for Landau levels

Based on a pair of representations obtained for Lie algebra h₄, the Hilbert space corresponding to all quantum states of Landau levels is split into an infinite direct sum of infinite-dimensional Hilbert subspaces. For any one of the Hilbert subspaces, we get linear combinations of their bases as generalized coherent states—the so-called Klauder–Perelomov and Gazeau–Klauder.