Topological properties of unbounded bicommutants

Mathot, Françoise

doi:10.1063/1.526510

Cited by 14 publications

(13 citation statements)

References 9 publications

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“…£ s ;ts*). We recall that J^f(^, Jf) is complete for £ s *, but not in general for t w or t s [24]. The locally convex topology on 901 generated by the family of seminorms on…”

Section: ®<=D(x\*) and X\*@ CI D(x$)mentioning

confidence: 99%

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Partial *-Algebras of Closable Operators. I. The Basic Theory and the Abelian Case

Antoine¹,

Inoue²,

Trapani³

1990

Publ. Res. Inst. Math. Sci.

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This paper, the first of two, is devoted to a systematic study of partial *-algebras of closable operators in a Hilbert space (partial Op*-algebras). After setting up the basic definitions, we describe canonical extensions of partial Op*-algebras by closure and introduce a new bounded commutant, called quasi-weak. We initiate a theory of abelian partial *-algebras.As an application, we analyze thoroughly the partial Op*-algebras generated by a single closed symmetric operator. §1. Introduction Ever since the pioneering days of Heisenberg's Matrix Mechanics, operator algebras have played a prominent role in quantum theories. For instance, the algebraic language is by now standard in quantum statistical mechanics (e. g. see the monograph of Bratteli and Robinson [1]). However the algebras used in this context consist invariably of bounded operators (in particular representations of abstract C*-algebras) and their bicommutants, that is, von Neumann algebras. In particular the latter play a crucial role in the Tomita-Takesaki theory [1].Yet this framework is often too narrow for applications. The next step is to consider algebras of unbounded operators, consisting of operators with a common dense invariant domain. Take for instance a nonrelativistic one particle quantum system. In the Schrodinger representation, with Hilbert space L 2 (R 3 ), the canonical variables are represented by the operators q and p, both unbounded and obeying the canonical commutation relations [p j? q k~] = id jk . The natural domain associated to this system is of course Schwartz space ^(R 3 ), and the corresponding *-algebra ^(£f) consists of all operators A such that A£f c £f and A*tf a

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“…£ s ;ts*). We recall that J^f(^, Jf) is complete for £ s *, but not in general for t w or t s [24]. The locally convex topology on 901 generated by the family of seminorms on…”

Section: ®<=D(x\*) and X\*@ CI D(x$)mentioning

confidence: 99%

“…To get a similar statement for an algebra of unbounded operators, a fortiori for a partial Op*-algebra 9K, we have to choose first the type of commutant, necessarily unbounded. Several possibilities are at our disposal [9], for instance the weak unbounded commutant [23,24] …”

Section: ®<=D(x\*) and X\*@ CI D(x$)mentioning

confidence: 99%

Partial *-Algebras of Closable Operators. I. The Basic Theory and the Abelian Case

Antoine¹,

Inoue²,

Trapani³

1990

Publ. Res. Inst. Math. Sci.

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“…Further, for the above statements (2) and (3) where ^T(^, $) is the set of all linear operators X in $ such that & (X) n&(X*)!3&. The study of unbounded commutants has been treated in [5,7,8,10,16,21], in particular Mathot has investigated topological properties of unbounded commutants of O| -algebras [21]. We have the following We introduce the notion of EW*-algebras which is another unbounded generalization of von Neumann algebras investigated by [9,12,14].…”

Section: In Order To Generalize the Notion Of Von Neumann Algebras Tomentioning

confidence: 99%

An Unbounded Generalization of the Tomita-Takesaki Theory

Inoue¹

1986

Publ. Res. Inst. Math. Sci.

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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 y 0 (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.

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“…It is possible to consider various topologies on a 4^-invariant set 0 Here we shall consider the so-called strong ^-topology (shortly s*-topology) [24] which is denned by the following set of semi-norms: …”

Section: 2 0 Topology On 21mentioning

confidence: 99%

“…Here, we can actually consider a weaker topology, the stronĝ -topology ( [23] for bounded operators, [24] for unbounded) which is a particular case of a quasi-uniform topology. In practice, when we shall consider representations of abstract partial ^-algebras by closed operators, the assumption of separability in the strong *-topology will come from the fact that we shall consider strongly continuous representations.…”

Section: Construction Of a Common Domain @ (X) For A Countable Set Ofmentioning

confidence: 99%