Differentiable absorption of Hilbert C*-modules, connections and lifts of unbounded operators

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Resumé

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C∗C∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C∗C∗-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C∗C∗-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correpondences) on Hilbert C∗C∗-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert C∗C∗-module.
OriginalsprogEngelsk
TidsskriftJournal of Noncommutative Geometry
Vol/bind11
Udgave nummer3
Sider (fra-til)1037-1068
ISSN1661-6952
DOI
StatusUdgivet - 2017

Fingeraftryk

Hilbert Modules
Hilbert C*-module
Unbounded Operators
Differentiable
Absorption
Theorem
Algebra
Scalar, inner or dot product
Compatibility
Stabilization
Isomorphic
Generator
Module

Citer dette

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abstract = "The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C∗C∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C∗C∗-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C∗C∗-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correpondences) on Hilbert C∗C∗-modules. These connections can in turn be used to define selfadjoint and regular {"}lifts{"} of unbounded operators which act on an auxiliary Hilbert C∗C∗-module.",
keywords = "Hilbert C∗C∗-modules, derivations, differentiable absorption, Grassmann connections, regular unbounded operators",
author = "Jens Kaad",
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Differentiable absorption of Hilbert C*-modules, connections and lifts of unbounded operators. / Kaad, Jens.

I: Journal of Noncommutative Geometry, Bind 11, Nr. 3, 2017, s. 1037-1068.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

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AU - Kaad, Jens

PY - 2017

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AB - The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C∗C∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C∗C∗-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C∗C∗-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correpondences) on Hilbert C∗C∗-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert C∗C∗-module.

KW - Hilbert C∗C∗-modules

KW - derivations

KW - differentiable absorption

KW - Grassmann connections

KW - regular unbounded operators

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