Algebras Of Unbounded Operators Research Articles

Strongly Commuting Selfadjoint Operators and Commutants of Unbounded Operator Algebras

Strongly Commuting Selfadjoint Operators and Commutants of Unbounded Operator Algebras

Strongly commuting selfadjoint operators and commutants of unbounded operator algebras

Let A 1 {A_1} and A 2 {A_2} be (unbounded) selfadjoint operators on a Hilbert space H \mathcal {H} which commute on a dense linear subspace of H \mathcal {H} . To conclude that A 1 {A_1} and A 2 {A_2} strongly commute, additional assumptions are necessary. Two propositions which contain such additional conditions are proved in §1. In §2 we define different commutants of unbounded operator algebras (form commutant, weak unbounded commutant, strong unbounded commutant) and we discuss the relations between them and their bounded parts. In §3 we construct a selfadjoint ∗ {*} -representation of the polynomial algebra in two variables for which the form commutant is different from the weak unbounded commutant.

Algebras of unbounded scalar-type spectral operators

If P: ? -* J?f (X) is a closed spectral measure in the quasicomplete locally convex space X and if T is a densely defined linear operator in X with domain invariant under each operator of the form /? fdP, with / a complex bounded ?-measurable function then T is closable and there exists a complex ?-measurable function / such that the closure of T is the spectral integral /? fdP if and only if T leaves invariant each closed subspace of X which is invariant under the range of the spectral measure P.

An Unbounded Generalization of the Tomita-Takesaki Theory II

Algebras of unbounded operators have been studying by many mathematicians (Borchers, Uhlmann, Lassner, Powers, Schmiidgen, Antoine, Gudder, etc... .) from situations of the physical applications as well as the sheer mathematical interest. The study of oneparameter automorphism groups and dynamics in unbounded operator algebras seems to be hardly done except [8, 17], It is well known that the Tomita-Takesaki theory plays an important role for such a study in von Neumann algebras. In this direction we consider an unbounded generalization of the Tomita-Takesaki theory, and treat modular automorphism groups of such algebras. We define the notion of unbounded left Hilbert algebras which is an unbounded generalization of left Hilbert algebras in the sense that the left multiplication is not necessarily bounded. Then a bicommutant 2T of an unbounded left Hilbert algebra 21 is defined and becomes an achieved left Hilbert algebra, and so it induces the fundamental theorem of Tomita for the left von Neumann algebra ^0(2T) and the right von Neumann algebra y0(2T) of 2T: /'^oCSO/'^^oCSO, J""^o(aO^~'''=*o(2r) for all t^R, where f is the modular conjugation operator of 2T and A" is the modular operator of ST. The first purpose is to extend the above results to an unbounded left Hilbert algebra 2L The following question arises.

Commutants and bicommutants of algebras of unbounded operators

The first purpose of this paper is to show that for each Op*-algebra (ℳ,𝒟) whose weak commutant ℳ′w is an algebra, there exists a closed Op*-algebra (ℳ̂,𝒟̂), which is the smallest extension of (ℳ,𝒟) satisfying ℳ̂w =ℳ′w and ℳ̂w 𝒟̂ =𝒟̂. The second purpose is to characterize an unbounded bicommutant ℳ″wσ of an Op*-algebra ℳ. The third purpose is to generalize the well-known Radon–Nikodym theorem for von Neumann algebras to Op*-algebras ℳ satisfying the von Neumann density type theorem ℳ̄ t*s =ℳ″wσ.

Topological Realizations of Calkin Algebras on Frechet Domains of Unbounded Operator Algebras

Let $\mathcal D$ be a dense linear subspace of a separable Hilbert space and let $\mathcal L^+(\mathcal D)$ be the maximal Op\*-algebra on $\mathcal D$ endowed with the uniform topology $\tau\_{\mathcal D}$. Suppose $\mathcal D$ is a Frechet space with respect to the graph topology of $\mathcal L^+(\mathcal D)$. Let $\mathcal C (\mathcal D)$ denote the two-sided \*-ideal of all operators in $\mathcal L^+(\mathcal D)$ which map bounded subsets of $\mathcal D$ into relatively compact subsets. We study the question of when the quotient algebra $\mathcal A (\mathcal D) \colon = \mathcal L^+(\mathcal D) / \mathcal C (\mathcal D)$, endowed with the quotient topology, has a topological realization as an Op\*-algebra.

Joint spectrum and unbounded operator algebras

Joint spectrum and unbounded operator algebras

An Unbounded Generalization of the Tomita-Takesaki Theory

Algebras of unbounded operators have been studying by many mathematicians (Borchers, Uhlmann, Lassner, Powers, Schmiidgen, Antoine, Gudder, etc... .) from situations of the physical applications as well as the sheer mathematical interest. The study of oneparameter automorphism groups and dynamics in unbounded operator algebras seems to be hardly done except [8, 17], It is well known that the Tomita-Takesaki theory plays an important role for such a study in von Neumann algebras. In this direction we consider an unbounded generalization of the Tomita-Takesaki theory, and treat modular automorphism groups of such algebras. We define the notion of unbounded left Hilbert algebras which is an unbounded generalization of left Hilbert algebras in the sense that the left multiplication is not necessarily bounded. Then a bicommutant 2T of an unbounded left Hilbert algebra 21 is defined and becomes an achieved left Hilbert algebra, and so it induces the fundamental theorem of Tomita for the left von Neumann algebra ^0(2T) and the right von Neumann algebra y0(2T) of 2T: /'^oCSO/'^^oCSO, J^o(aO^~'''=*o(2r) for all t^R, where f is the modular conjugation operator of 2T and A is the modular operator of ST. The first purpose is to extend the above results to an unbounded left Hilbert algebra 2L The following question arises.

When is a Field a Jacobi-Field? A Characterization of States on Tensor Algebras

In order to analyse the positivity condition for states on the tensoralgebra E over certain (function-)spaces E more efficiently a representation of the components of the states in terms of a set of independent parameters is suggested. For this purpose the concept of a Jacobifield is introduced. In the case of a finite dimensional space E every state on E is parametrized this way. If however the basic space E is infinite dimensional additional domain problems arise related to algebras of unbounded operators which are involved naturally. It is analysed to which extent this parametrization in terms of Jacobi-fields also works in the general case, and it is shown that for many basic spaces E which occur in applications most of the states admit indeed such a parametrizati on. This then also means a corresponding decomposition for the associated algebra of unbounded operators into independent components. Several applications are indicated.

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L+(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L+(D) is well-ordered by inclusion and the α-th closed ideal 𝔍α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then 𝔍α. An element A in L+(D) belongs to the minimal closed ideal 𝔍0 if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.

Some algebras of unbounded operators in an indefinite inner product vector space

A general theory of unbounded representations of *-algebras in the Krein space is formulated. Following the corresponding theory of unbounded representations in the Hilbert space, the connection between states and representations, properties of covariant representations, the notion of irreducibility, and decomposition into irreducible parts are discussed.

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB*-algebra leads to the study of properGB*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC*-algebra and two other representation theorems forb*-algebras (also calledlmc*-algebras), one representinga b*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeLp-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andLw-integral of a measurable field ofC*-algebras are discussed briefly.

Spatial theory for algebras of unbounded operators

In this paper, the spatial theory of ∗-automorphisms is investigated in the context of algebras of unbounded operators. In particular, it is shown that ∗-automorphisms, satisfying some order relation, of the Op ∗-algebra generated by the position and momentum operators q j , p j ( j=1,...., n) on the Schwartz space I( R n) are unitarily implemented.

Topological properties of unbounded bicommutants

We consider different possible definitions of unbounded commutants and unbounded bicommutants of a set or an algebra of unbounded operators. We investigate their behavior with respect to various topologies. In particular we give sufficient conditions in order that bicommutants be the closure of the original set of operators with respect to some of those topologies. We investigate some special classes of algebras (symmetric, self-adjoint, regular, V* algebras) for which several or all of the bicommutants coincide and are the closure of the algebra with respect to some or all of the considered topologies.

Joint numerical ranges for unbounded normal operators

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

Irreducible representations of a class of unbounded operator algebras

Irreducible representations of a class of unbounded operator algebras

Algebras of unbounded operators and vacuum superselection rules in quantum field theory

Algebras of unbounded operators and vacuum superselection rules in quantum field theory

V*-algebras: A particular class of unbounded operator algebras

We consider a weak unbounded commutant for a set of unbounded operators and we examine op*-algebras which coincide with some of their bicommutant. This class of op*-algebras, called V*-algebras, shows some properties close to those which hold true for bounded operator algebras.

Closed linear operators with domain containing their range

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op* algebras and vector states on them

Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op* algebras and vector states on them