Hilbert 𝐶*-Modules

Manuilov, Vladimir; Troitsky, Evgenij

doi:10.1090/mmono/226

Cited by 142 publications

(155 citation statements)

References 4 publications

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“…The definition below amounts essentially to replacing M ∞ (A) with A⊗K in the definition above (this is a theorem), and also gives a new and very useful characterisation of Cuntz subequivalence. We refer the reader to [38] and [43] for background material on Hilbert C * -modules. Consider A as a (right) Hilbert C * -module over itself, and let H A denote the countably infinite direct sum of copies of this module.…”

Section: The Elliott Invariant and The Original Conjecture The Elliomentioning

confidence: 99%

Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

145

137

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Abstract. We report on recent progress in the program to classify separable amenable C * -algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

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Section: The Elliott Invariant and The Original Conjecture The Elliomentioning

confidence: 99%

Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

145

137

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“…Thus 20) where h x = δ x h ∈ F 0 is the function that takes the value h at x and is null elsewhere. Equivalently,…”

Section: Proof (1)⇒(2)mentioning

confidence: 99%

Representations of ⁎-semigroups associated to invariant kernels with values adjointable operators

Gheondea

2015

Linear Algebra and its Applications

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We consider positive semidefinite kernels valued in the * -algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of * -semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the * -semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non-topological level.

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“…For facts on Hilbert C * -modules we refer the reader to [16,17,22]. We recall here just the most important for us statements.…”

Section: Branched Coveringsmentioning

confidence: 99%

Quantization of branched coverings

Pavlov¹,

Troitskii²

2011

Russ. J. Math. Phys.

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Abstract. We identify branched coverings (continuous open surjections p : Y → X of Hausdorff spaces with uniformly bounded number of pre-images) with Hilbert C*-modules C(Y ) over C(X) and with faithful unital positive conditional expectations E : C(Y ) → C(X) topologically of index-finite type. The case of non-branched coverings corresponds to projective finitely generated modules and expectations (algebraically) of index-finite type. This allows to define non-commutative analogues of (branched) coverings.

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Hilbert 𝐶*-Modules

Abstract: We use essential ideals in a C * -algebra A to extend the inner product on a Hilbert C * -module M to a greater C * -submodule of the dual Banach module M ′ . This extension may go beyond the second dual module M ′′ .

Cited by 142 publications

References 4 publications

Regularity properties in the classification program for separable amenable C*-algebras

Regularity properties in the classification program for separable amenable C*-algebras

Representations of ⁎-semigroups associated to invariant kernels with values adjointable operators

Quantization of branched coverings

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