A property of Hilbert modules and Fredholm operators over 𝐶*-algebras

Notes on twisted equivariant K-theory for C*-algebras

International Journal of Mathematics ◽

10.1142/s0129167x16500580 ◽

2016 ◽

Vol 27 (06) ◽

pp. 1650058 ◽

Cited By ~ 7

Author(s):

Yosuke Kubota

Keyword(s):

Topological Insulators ◽

Banach Algebras ◽

Hilbert Modules ◽

Fredholm Operators ◽

C Algebras ◽

K Theory

In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed–Moore for [Formula: see text]-graded [Formula: see text]-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele’s K-theory for [Formula: see text]-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the [Formula: see text]-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

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Global Perturbation of Nonlinear Eigenvalues

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2127 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Julián López-Gómez ◽

Juan Carlos Sampedro

Keyword(s):

Perturbation Theory ◽

Spectral Parameter ◽

Classical Theory ◽

General Setting ◽

Perturbation Parameter ◽

Linear Operators ◽

Fredholm Operators ◽

Index Zero ◽

Finite Dimensional ◽

Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

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Economical finite rank perturbations of semi-Fredholm operators

Mathematische Zeitschrift ◽

10.1007/bf01184676 ◽

1988 ◽

Vol 198 (3) ◽

pp. 431-434 ◽

Cited By ~ 3

Author(s):

M�che�l � Searc�id

Keyword(s):

Finite Rank ◽

Fredholm Operators

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Fredholm operators on locally compact spaces

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf01762543 ◽

1984 ◽

Vol 138 (1) ◽

pp. 191-210

Author(s):

N. Teleman

Keyword(s):

Fredholm Operators ◽

Locally Compact ◽

Compact Spaces ◽

Locally Compact Spaces

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On a new topology in the space of Fredholm operators

Annals of Global Analysis and Geometry ◽

10.1007/bf00127860 ◽

1989 ◽

Vol 7 (2) ◽

pp. 93-106 ◽

Cited By ~ 4

Author(s):

Anvar Irmatov

Keyword(s):

Fredholm Operators

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Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

Glasnik Matematicki ◽

10.3336/gm.56.2.08 ◽

2021 ◽

Vol 56 (2) ◽

pp. 343-374

Author(s):

Boris Guljaš ◽

Keyword(s):

Hilbert Space ◽

Hilbert Modules ◽

Strict Topology ◽

Direct Sums ◽

Essential Ideal ◽

Closed Submodule ◽

Closed Submodules ◽

Hereditary Algebras ◽

Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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