Algebras of unbounded operators and quantum dynamics

Lassner, G.

doi:10.1016/0378-4371(84)90263-2

Cited by 102 publications

(63 citation statements)

References 8 publications

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“…\ such that (gi(X)} is dense in 3t? A (X) and similarly a dense set {fj in jf At such that (fi(X)} is dense in Jf A *(X) 9 we get that for any f…”

Section: \(F(x) A(x)g(x» =\(A*(x)f(x)mentioning

confidence: 99%

“…For instance, there exist some quantum statistical systems for which the thermodynamical limit does not exist in a C*-topology [8] but only in the completion of some Op*-algebra. This completion, called a quasi-algebra [9], is no longer an algebra itself, but a more general structure where the product between two elements need not be defined.…”

Section: Introductionmentioning

confidence: 99%

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Integral Decomposition of Partial *-Algebras of Closed Operators

Mathot¹

1986

Publ. Res. Inst. Math. Sci.

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“…\ such that (gi(X)} is dense in 3t? A (X) and similarly a dense set {fj in jf At such that (fi(X)} is dense in Jf A *(X) 9 we get that for any f…”

Section: \(F(x) A(x)g(x» =\(A*(x)f(x)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integral Decomposition of Partial *-Algebras of Closed Operators

Mathot¹

1986

Publ. Res. Inst. Math. Sci.

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“…As is clear, the multiplication of a test function times a tempered distribution, makes of (S (R n ), S(R n )) a quasi*-algebra in the sense of Lassner [5,6] but, in this set-up, the corresponding lattice of multipliers is rather trivial. For this reason, moving within the framework of the so-called duality method [1, Sect.…”

Section: §1 Introductionmentioning

confidence: 99%

“…As is clear, the multiplication of a test function times a tempered distribution, makes of (S (R n ), S(R n )) a quasi*-algebra in the sense of Lassner [5,6] …”

Section: Introductionmentioning

confidence: 99%

Partial *-algebras of Distributions

Trapani¹,

Tschinke²

2005

Publ. Res. Inst. Math. Sci.

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The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of C ∞ -vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an abstract version of the Bessel potential spaces, are used to factorize the product. §1. Introduction Distributions are, as is well-known, typical objects that can be multiplied only if some very particular circumstances occur. Nevertheless, products of distributions, sometimes understood only in formal sense, frequently appear in physical applications (for instance in quantum field theory) and play a relevant role in the theory of partial differential equations. For these reasons many possibilities of defining a (partial) multiplication have been suggested in the literature (see [1] for an overview) dating back to the famous Schwartz paper devoted to the impossibility of multiplying two Dirac delta measures massed at the same point [2].Reconsidering an idea developed in [3], we study in this paper the possibility of making of the space S (R n ) a partial *-algebra in the sense of [4]. As is clear, the multiplication of a test function times a tempered distribution, makes of (S (R n ), S(R n )) a quasi*-algebra in the sense of Lassner [5,6] Given a rigged Hilbert space, D ⊂ H ⊂ D , we denote with L(D, D ) the set of all continuous linear maps fromD) (for this reason we denote with † both involutions).uous. Hence, the problem of multiplying two distributions U, V ∈ S (R n ) can be viewed in terms of multiplication of the corresponding operators, S (R n )) and then investigating conditions under which L U ·L V = L W for some W ∈ S (R n ). This was partially done in [3] also to a certain degree of abstractness: therein, in fact, tempered distributions were considered as a special case of the so called A-distributions defined as elements of the (conjugate)Partial *-algebras of Distributions

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“…But often limits of local observables may fail to exist in the C*-norm of the local algebra and completions under weaker topologies (that, in general, fail to be *-algebras) must be taken. This puts on the stage partial *-algebras [1] and, more precisely, quasi *-algebras, originally introduced by Lassner for the study of the thermodynamical limits of certain spin systems [10,11].…”

Section: Introductionmentioning

confidence: 99%