Self-Duality and $C*$-Reflexivity of Hilbert $C*$-Moduli

Frank, Michael

doi:10.4171/zaa/390

Cited by 54 publications

(54 citation statements)

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“…The extension of bounded A-linear operators from M to N is continuous with respect to the w * -topology on N . For topological characterizations of self-duality of Hilbert C * -modules over W * -algebras we refer to [19], [8,Thm. 3.2] and to [21,22]: a Hilbert C * -module K over a W * -algebra B is self-dual, if and only if its unit ball is complete with respect to the topology induced by the semi-norms {|f ( ., x )|:…”

Section: Orthogonality-preserving Mappingsmentioning

confidence: 99%

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Frank¹,

Mishchenko²,

Pavlov³

2011

Journal of Functional Analysis

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We investigate orthonormality-preserving, C * -conformal and conformal module mappings on full Hilbert C * -modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C * -algebra of coefficients combined with an isometric module operator as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfills the equality T (x), T (y) = |λ| 2 x, y for any elements x, y of the Hilbert C * -module. At the contrary, C * -conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators.

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Section: Orthogonality-preserving Mappingsmentioning

confidence: 99%

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Frank¹,

Mishchenko²,

Pavlov³

2011

Journal of Functional Analysis

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“…Now let 9I be a commutative AW*-algebra, {X, <(, >} be a Hilbert 91-module and I be a directed net. A norm-bounded set {x,: aE I} of elements of X has a limit x in -y with respect to the topology T if and only if the limit These topologies are investigated in [6]. They have the following properties: (i) The Tl-limit (the T2-limit) x -y is unique.…”

Section: Q2(x) [R] = R(x)mentioning

confidence: 99%

Elements of Tomita-Takesaki theory for embeddable AW*-algebras

Frank

1989

Ann Glob Anal Geom

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In this paper a Tomita-Takesaki type theorem for embeddable AW*-algebras M1 possessing in one of their "normal" representations on a faithful self-dual Hilbert 3(/)-module * a cyclic-separating element x E f is proved. The proving techniques are derived mainly from a paper of M. A. Rieffel and A. van Daele with some changes and generalizations. The crucial role in the proof play some special real subspaces oX of f generated by the self-adjoint part of T1~ and by x E #f, and operators related to X. The results from their investigation are of more general interest. § 1. IntroductionIn this paper we prove a Tomita-Takesaki type theorem for embeddable AW*-algebras possessing in one of their "normal" representations a cyclic-separating element. For this aim we use the way of proving determined in a paper of M. A. Rieffel and A. van Daele ([16]) with some changes in the proving techniques.The organization of the present paper is as follows. The second paragraph is concerned with basic definitions and with facts related to the subject. Under the supposition that X' is a self-dual Hilbert WI-module over a certain commutative AW*-algebra I we consider some special real self-dual Hilbert ¶Wh-submoduli X of ' and some bounded operators arising from them in the third paragraph. In particular, a conjugate-91-linear involution J on X with respect to oX is defined. In paragraph four order continuous unitary one-parameter groups {/izt:t E R} are investigated being defined for each such real subspace o-of Xf on Xf. The relation of those groups to a generalized K.M.S. condition is established. (Similar results are contained in a preprint of H. Halpern ([7], private communication)). We remark that the structure of the paragraphs three and four was suggested by [16]. Paragraph five is concerned with the Tomita-Takesaki type theorem for embeddable AW*-algebras possessing in one of their "normal" representations a cyclic-separating element. A interpretation of the results of the former paragraphs in terms of locally trivial Hilbert bundles over stoneian compact spaces is given in paragraph six. § 2. PreliminariesIn this part of the present paper the restriction on 91 to be commutative is often superfluous. But it will be necessary in further discussions.

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“…Unfortunately, for every infinite-dimensional C*-algebra A of coefficients the standard countably generated Hilbert A-module l 2 .A/ D A˝l 2 fails to be self-dual ( [3]). So the extension of key results of the theory of Hilbert spaces to entire classes of Hilbert A-modules over certain fixed C*-algebras A did not seem to be attainable beyond finite-dimensional matrix algebras A. G. G. Kasparov's approach to K-theory and the results of W. L. Paschke pointed to finitely and countably generated Hilbert C*-modules over unital C*-algebras and to self-dual Hilbert W*-modules over W*-algebras as the best available candidates for generalizing key parts of Hilbert space theory, cf.…”

Section: Introductionmentioning

confidence: 99%

Characterizing C-algebras of compact operators by generic categorical properties of Hilbert C-modules

Frank

2008

J. K-Theory

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C*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

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Self-Duality and $C$-Reflexivity of Hilbert $C$-Moduli

Cited by 54 publications

References 4 publications

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Elements of Tomita-Takesaki theory for embeddable AW*-algebras

Characterizing C-algebras of compact operators by generic categorical properties of Hilbert C-modules

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Self-Duality and $C*$-Reflexivity of Hilbert $C*$-Moduli

Cited by 54 publications

References 4 publications

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Elements of Tomita-Takesaki theory for embeddable AW*-algebras

Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules

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Self-Duality and $C$-Reflexivity of Hilbert $C$-Moduli

Characterizing C-algebras of compact operators by generic categorical properties of Hilbert C-modules