Quantum Fields as Pointlike Localized Objects

We show that Wightman fields are always well defined objects at each space-time point in the meaning of sesquilinear forms. These sesquilinear forms “sandwiched” with e^–cH (H is the energy operator) are closable operators. We use these results to extend some assertions of Fredenhagen and Hertel about the recovering of Wightman fields from a Haag -Kastler theory of local observables.     

On Jaffe Fields at a Point and Algebras of Local Observables

It is shown that JAFFE fields exist at each space-time point in the sense of sesquilinear forms. The connection between JAFFE fields and algebras of local observables is investigated. An example of a proper JAFFE field is given.     

Modular Sesquilinear Forms and Generalized Stinespring Representation

We consider completely positive maps defined on locally C*-algebra and taking values in the space of sesquilinear forms on Hilbert C*-module M. We construct the Stinespring type representation for this type of maps and show that any two minimal Stinespring representations are unitarily equivalent.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Dualities for infinite-dimensional projective geometries

As main theorem we show that any duality in the projective geometry associated to a vector space is induced by a non-degenerate sesquilinear form on this space. In particular, any polarity is induced by a non-degenerate orthosymmetric sesquilinear form.Supported by a grant from the Swiss National Founds for Scientific Research.     

Error Estimates for Discretized Quantum Stochastic Differential Inclusions

Abstract

This paper is concerned with the error estimates involved in the solution of a discrete approximation of a quantum stochastic differential inclusion (QSDI). Our main results rely on certain properties of the averaged modulus of continuity for multivalued sesquilinear forms associated with QSDI. We obtained results concerning the estimates of the Hausdorff distance between the set of solutions of the QSDI and the set of solutions of its discrete approximation. This extends the results of Dontchev and Farkhi Dontchev, A.L.; Farkhi, E.M. (Error estimates for discre‐ tized differential inclusions. Computing 1989, 41, 349–358) concerning classical differential inclusions to the present noncommutative quantum setting involving inclusions in certain locally convex space.     

On the definition of Hilbert space

The projection theorem expresses a central feature of classical Hilbert space. Do other infinite dimensional sesquilinear spaces share this property? We show here that this is not the case for several prominent candidates; in particular Kalish's p-adic Hilbert spaces, Springers non archimedean normed spaces, the positive definite spaces over ordered fields. This yields interesting characterizations of classical Hilbert space.     

The Kato Conjecture for Elliptic Differential-Difference Operators with Degeneration in a Cylinder

Second-order elliptic differential-difference operators with degeneration in a cylinder associated with closed densely defined sectorial sesquilinear forms in L₂(Q) are considered. These operators are proved to satisfy the Kato conjecture on the square root of an operator.     

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for

(i)

bilinear forms over an algebraically closed or real closed field;

(ii)

sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and

(iii)

sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F; the canonical matrices are based on any given set of canonical matrices for similarity over F.

A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481-501].     

A Two-Parameter Quantum (2+1)-Superspace and its Deformed Derivation Algebra as Hopf Superalgebra

As is well known, a Hopf algebra setting is an efficient tool to study some geometric structures such as the Maurer-Cartan invariant forms and the corresponding vector fields on a noncommutative space. In this study we introduce a two-parameter quantum (2+1)-superspace with a Hopf superalgebra structure.We also define some derivation operators acting on this quantum superspace, and we show that the algebra of these derivations is a Hopf superalgebra. Furthermore it will be shown how the derivation operators lead to a bicovariant differential calculus on the two- parameter quantum (2+1)-superspace. In conclusion, based on the bicovariant differential calculus, the Maurer-Cartan right invariant differential forms and the corresponding quantum Lie superalgebra are given.     

Representable Linear Functionals on Partial *-Algebras

A GNS-like *-representation of a partial *-algebra \mathfrak A{{\mathfrak A}} defined by certain representable linear functionals on \mathfrak A{{\mathfrak A}} is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.     

Diagonalization and representation results for nonpositive sesquilinear form measures

We study decompositions of operator measures and more general sesquilinear form measures E into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent E as a trace class valued measure of bounded variation on a new Hilbert space related to E. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals.     

The convexity of the range of three Hermitian forms and of the numerical range of sesquilinear forms

This note contains a demonstration of the convexity of the joint range of three Hermitian forms, and, as an immediate corollary, of the convexity of the numerical range of arbitrary sesquilinear forms.     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

C*代数l上的左模的半双线性型的稳定性

本文分别研究了 C^*-代数l上的单位左模和具有实阶零性质的CC^*代数l上的单位左模的半双线性型的 Hyers-Ulam 稳定性.     

On the structure of a cone of normal unbounded completely positive maps

The paper suggests a constructive characterization of unbounded completely positive maps introduced earlier by Chebotarev for the theory of quantum dynamical semigroups. We prove that such cones are generated by a positive self-adjoint “reference” operator ΛεB(H) as follows: for any completely positive unbounded map Ф(·)εCPn*(F) these exists a completely positive normal bounded mapR(·)εCPn(H) such that ϕ(·)=ΛR(·)Λ. The class contains mappings that are unclosable sesquilinear forms. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 194–205, February, 1999.     

Complexity of the satisfiability problem for multilinear forms over a finite field

Multilinear forms over finite fields are considered. Multilinear forms over a field are products in which each factor is the sum of variables or elements of this field. Each multilinear form defines a function over this field. A multilinear form is called satisfiable if it represents a nonzero function. We show the N P-completeness of the satisfiability recognition problem for multilinear forms over each finite field of q elements for q ≥ 3. A theorem is proved that distinguishes cases of polynomiality and NP-completeness of the satisfiability recognition problem for multilinear fields for each possible q ≥ 3.     

Closedness and lower semicontinuity of positive sesquilinear forms

The relationship between the notion of closedness, lower semicontinuity and completeness (of a quotient) of the domain of a positive sesquilinear form defined on a subspace of a topological vector space is investigated and sufficient conditions for their equivalence are given.      

A generalization of the original Jordan–von Neumann theorem

The problem of representability of quadratic functionals (acting on modules over unital complex ∗-algebras), by sesquilinear forms, is generalized by weakening the homogeneity equation. The corresponding representation theorem can be considered as a generalization of (the original form of) the classical Jordan–von Neumann characterization of complex inner product spaces.