Soler's Theorem and Characterization of Inner Product Spaces

Using Soler's result, we show that the existenceof at least one finitely additive probability measure onthe system of all orthogonally closed subspaces of Swhich is concentrated on a one-dimensional subspace of E can imply that E is a real,complex, or quaternionic Hilbert space. In addition,using the concept of test spaces of Foulis and Randalland introducing various systems of subspaces of E , we give some characterizations of inner productspaces which imply that E is a real, complex, orquaternionic Hilbert space.     

Gleason's Theorem in W*J-Algebras in Spaces with Indefinite Metric

We characterize measures on hyperbolic logicsassociated to von Neumann algebras acting in a spacewith an indefinite metric. An analog to the Gleasontheorem is proved.     

A New Algebraic Criterion for Completeness of Inner Product Spaces

We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M = S holds, satisfies the -Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a -complete lattice, and that of the author who required E(S) to be a -orthocomplete OMP.     

On GNS Representations¶on Inner Product Spaces

A generalization of the GNS construction to hermitian linear functionals W defined on a unital *-algebra is considered. Along these lines, a continuity condition (H) upon W is introduced such that (H) proves to be necessary and sufficient for the existence of a J-representation on a Krein space . The property whether or not the Gram operator J leaves the (common and invariant) domain of the representation invariant is characterized as well by properties of the functional W as by those of . Furthermore, the interesting class of positively dominated functionals is introduced and investigated. Some applications to tensor algebras are finally discussed. Received: 21 May 1996 / Accepted: 12 May 1997     

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.     

On the Relation Between Time Representations and Inner Product Spaces

Stable states (particles), ghosts and unstable states (particles) come with different types of time representations in unitary groups—definite or indefinite. These representations are discussed with respect to the induced inner product spaces as extensions of Hilbert spaces. Unstable particles with their decay channels are treated as higher dimensional probability collectives.     

States on Subspaces of Inner Product Spaces with the Gleason Property

We show that the range of every finitely additive state on the system of all orthogonally closed subspaces of an infinite-dimensional inner product space E satisfying the Gleason property is equal to the real interval [0, 1]. Every pre-Hilbert space satisfies the Gleason property, and in Keller spaces it fails to hold.     

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.     

On One Possible Generalization of the Regression Theorem

A general approach to derivation of formally exact closed time-local or time-nonlocal evolution equations for non-equilibrium multi-time correlations functions made of observables of an open quantum system interacting simultaneously with external time-dependent classical fields and dissipative environment is discussed. The approach allows for the subsequent treatment of these equations within a perturbative scheme assuming that the system-environment interaction is weak.     

A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda [HHM].     

Gleason-Type Theorem for Linear Spaces over the Field of Four Elements

We prove an analog of the famous Gleason theoremfor additive functions on the orthomodular poset of allprojections defined on an n-dimensional linear spaceover the field consisting of four elements. An essential part of the proof consists in acomputer calculation.     

On the Non-linear Generalization of the Gyarmati Principle and Theorem

The validity of the Gyarmati theorem has been extended to non-linear types of constitutive equations governing dissipative processes which have recently been suggested. It has been proved that by extending the validity of the theorem, the validity of Gyarmati's variational principle of thermodynamics is guaranted for non-linear theories adequate to constitutive equations in question as well. The theory was applied for stationary temperature distribution in a rigid bar.     

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that

A generalization of Wigner's R-matrix method

A generalization of the Wigner's non-relativistic R-matrix theory of scattering by a central potential field is proposed. The idea is to use an R-matrix expansion basis generated by a Sturm-Liouville problem with an eigenparameter included both in a differential equation and in a boundary condition (in the standard theory an R-matrix basis is obtained by solving an eigenvalue problem with fixed boundary conditions). A partial fraction expansion of an R^(η)-matrix introduced is derived and shown to converge faster than a partial fraction expansion of Wigner's R-matrix used in the standard theory.     

PCT Theorem for the Operator Product Expansion in Curved Spacetime

We consider the operator product expansion for quantum field theories on general analytic 4-dimensional curved spacetimes within an axiomatic framework. We prove under certain general, model-independent assumptions that such an expansion necessarily has to be invariant under a simultaneous reversal of parity, time, and charge (PCT) in the following sense: The coefficients in the expansion of a product of fields on a curved spacetime with a given choice of time and space orientation are equal (modulo complex conjugation) to the coefficients for the product of the corresponding charge conjugate fields on the spacetime with the opposite time and space orientation. We propose that this result should be viewed as a replacement of the usual PCT theorem in Minkowski spacetime, at least in as far as the algebraic structure of the quantum fields at short distances is concerned.     

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrödinger equations with nonlocal smooth nonlinearity are also given in this paper.     

Full Poncelet Theorem in Minkowski dS and AdS Spaces

We study the reflection of a straight line or a billiard on a plane in an n-dimensional Minkowski space. It is found that the reflection law coincides with that defined with respect to confocal quadratic surfaces in projective geometry. We then establish the full Poncelet theorem which holds in projective geometry in n-dimensional Minkowski space and in their quadratic surfaces including de Sitter and AdS spaces.     

An elementary proof for the non-bijective version of Wigner's theorem

The non-bijective version of Wigner's theorem states that a map which is defined on the set of self-adjoint, rank-one projections (or pure states) of a complex Hilbert space and which preserves the transition probability between any two elements, is induced by a linear or antilinear isometry. We present a completely new, elementary and very short proof of this famous theorem which is very important in quantum mechanics. We do not assume bijectivity of the mapping or separability of the underlying space like in many other proofs.     

A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions

Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S ² × S ¹. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).     

Composite S-Brane Solutions on the Product of Ricci-Flat Spaces

A family of generalized S-brane solutions with orthogonal intersection rules and n Ricci-flat factor spaces in the theory with several scalar fields and antisymmetric forms is considered. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. These subclasses contain sub-families of solutions with accelerated expansion of certain factor spaces. The solutions depend on charge densities of branes, their dimensions and intersections, dilatonic couplings and the number of dilatonic fields.