Topological aspects of algebras of unbounded operators

Arnal, Didier; Jurzak, J. P.

doi:10.1016/0022-1236(77)90066-0

Cited by 36 publications

(11 citation statements)

References 10 publications

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“…Let 2) be a dense linear subspace of a separable complex Hubert space X endowed with, the graph.topology / (see Section 1 for precise definitions). Suppose 2) [1] is a Frechet space. Let (2)) 1enote the set of all operators in (2)) which map each bounded subset of 2)[/] into a relatively compact subset of 2) [1].…”

Section: Introductionmentioning

confidence: 99%

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

Schmüdgen¹

1986

Z. Anal. Anwend.

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0 seiein dichtèr linearer Teilraum einés separablen Hulbertraumes und .)0+(2)) die maximale OpsAlgebra auf 2), versehen mit der gleichmaBigen Topologie . Wir sezen vorau8, daB 2) bezuglich der Graphtopologie von 2'+(2) em Frechetraum ist. Weiter sel (2)) das zweiseitige *-Ideal aller Operatoren aus . + (2)), die beschrankte Teilmengen' von 2) in relativ kompakte Teilmengen abbilden. Es wird untersucht, wann die Faktoralgebra 4(2)) := 1+(2))/e(2)), versehen mit der Faktorraumtopologie, eine topologische Realisierung als Op*-Algebra besitzt. ilycri, 2) njioiioe JuIHeftrloe n01p0cTpaHcTl30 cenapa(5eJlbHOro rHJIb6epTOBa [Ip0CTpaHCTBa ii [I3ICTh 1+(2)) MaKcnMaJIbuan Op*-aire6pa iiag 2), cHa6ëllHa g panHoMepliofl ronøy1or11eft rD . llpeJuloJIaraeTcH, 'ITO 2) HB.iHeTcn H0CT3HCTC0M (Dpewe OTHOCHTCJIbIIO ronojiorn no-pO3se1Hoft rpa4)-HopMaMH oneparopon 113 1+ (2)).. 54epe3 (2)) o603iia'iaerca AByCT0P0HHHft s-itéa;i Tex oneparopon 113 1(2)), KOTOPMC iiepeBOAFIT orpaiiueiiiie nOMHOHeCT8a npocTpaHcTBa 0 B orHocwreJIbHo 1-oMnaKTHbIe noLMlJocecTBa. l4CcJ1eyeTcH BO[1OC 0 TOM, iora 4aHTopaIre6pa íf(2)) := .2'+(2)/(2)) cHaölKenllan TOflO.'101'Hefl ())aXTOPHPOCTPaIICTBaOIIycFcaeT T0no1orn9eci-y10 peaiinaaiuiio HaK Op*-aire6pa. Let 2) be a dense linear subspace of a separable Hubert space and let . '+(2)) be the maximal Op*-algebra on 2) endowed with f he uniform topology r. Suppose 2) is a Frechet space with respect to the graph topology of 1+ (2)). Let (2)) denofe the two-sided *-ideal of all operators in 1+ (2)) which map bounded subsets of 2) into relatively compact subsets. We study the question of when the quotient algebra 4(2)) 1'+(2))/(2)), endowed with the quotient topology, has a topological realization as an Op.-algebra.31 Analysis Rd. 5, heft 6 (1986) ', .

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Section: Introductionmentioning

confidence: 99%

“…Suppose 2) [1] is a Frechet space. Let (2)) 1enote the set of all operators in (2)) which map each bounded subset of 2)[/] into a relatively compact subset of 2) [1]. Then (2) is a r-closed two-sided *-ideal of (2)) which contains the finite ra'nk operators in .…”

Section: Introductionmentioning

confidence: 99%

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

Schmüdgen¹

1986

Z. Anal. Anwend.

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“…in parallel with theory of von Neumann algebras as a continuation of [7]. (Also see [14].) The set B(&, <&) of all continuous sesquilinear forms on @ (the continuity relative to the collection of norms @ 3 x»-> ||^4x|[, A e 91) plays the role of the set L(j^) of all bounded linear operators on 3?…”

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confidence: 99%

On a Certain Class of *-Algebras of Unbounded Operators

Araki¹,

Jurzak²

1982

Publ. Res. Inst. Math. Sci.

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A *-algebra SI of linear operators with a common invariant dense domain & in a Hilbert space is studied relative to the order structure given by the cone Sl + of positive elements of SI (in the sense of positive sesquilinear form on &) and the p-topology defined as an inductive limit of the order norm p A (of the subspace SI X with A as its order unit) with A eSl + . In particular, for those SI with a countable cofinal sequence A, in ST + such that AT 1 eSI, the ,0-topology is proved to be order convex, any positive elements in the predual is shown to be a countable sum of vector states, and the bicommutant within the set B(&, &) of continuous sesquilinear forms on 3f is shown to be the ultraweak closure of SI. The structure of the commutant and the bicommutant are explicitly given in terms of their bounded operator elements which are von Neumann algebras and the commutant of each other. § 1. Introduction Our aim in this paper is to develop a theory of a certain class of *-algebras 91 of linear operators with a common invariant dense domain 2 in a Hilbert space 3? in parallel with theory of von Neumann algebras as a continuation of [7]. (Also see [14].) The set B(&, <&) of all continuous sesquilinear forms on @ (the continuity relative to the collection of norms @ 3 x»-> ||^4x|[, A e 91) plays the role of the set L(j^) of all bounded linear operators on 3? in theory of von Neumann algebras. For those 91 satisfying Condition I described below, we can give the decomposition of continuous linear forms into positive components, i.e. the strong normality of the positive cone 9l + , the description of positive elements in the predual of 91 and the notion of commutant, for which the bicommutant coincides with the ultraweak closure.Under weaker Condition I 0 or IQ, also described below, structure of the

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“…Thus, condition (2.1) is fulfilled and we have a -~ =zst.REMARK. In[1], Op *-algebras satisfying condition (1.1) are called hyperfinite.…”

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confidence: 99%

Two theorems about topologies on countably generatedOp*-algebras

Schmüdgen

1980

Acta Mathematica Academiae Scientiarum Hungaricae

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The present paper deals with the topologization of unbounded operator algebras (Op.-algebras) in Hilbert space. We consider two possible topologies, the so-called uniform topology ~a introduced in [2] and the strong operator topology a ~. We characterize the countably generated [closed] Op.-algebras ~r for which z~ [resp. a ~] agrees with the strongest locally convex topology on ~'. Our main theorems contain some known result for concrete Op.-algebras ([2], [3], [4], [6]).In [5], theorem 1 was used in proving that for certain Op,-algebras ~ (for example, the Op .-algebra of all differential operators with polynomial coefficients on the Schwartz space Se (R~)) all linear functionals f on ~r are trace functionals, i.e. they can be given by f(a)=Trta, aE~], t an appropriate nuclear operator.

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Topological aspects of algebras of unbounded operators

Cited by 36 publications

References 10 publications

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

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

On a Certain Class of *-Algebras of Unbounded Operators

Two theorems about topologies on countably generatedOp*-algebras

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