On Inductive Limits for Systems of C*-Algebras

Gumerov, R. N.; Lipacheva, E. V.; Grigoryan, T. A.

doi:10.3103/s1066369x18070083

Cited by 13 publications

(8 citation statements)

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“…For additional results in the theory of C * -algebras we refer the reader, for example, to [27]; [28,Ch. 4,§ 7] and [25]. Necessary facts from the theory of categories and functors are contained, for example, in [28, Ch.…”

Section: Preliminariesmentioning

confidence: 99%

“…We consider the topology on the index set I which was introduced in [7,8]. For the convenience of the reader we recall the definition of this topology and its properties.…”

Section: Topology On An Index Setmentioning

confidence: 99%

“…Examples of different topological spaces (I, τ ) are contained in [7,8]. In particular, (I, τ ) may be: a non-Hausdorff space [ Here we give an example of a compact space.…”

Section: Topology On An Index Setmentioning

confidence: 99%

“…A part of motivation for studying inductive systems of Toeplitz algebras comes from [6,7,8,9]. The paper [6] contains results on limit automorphisms for inductive sequences of Toeplitz algebras which are closely related to the facts on the mappings of topological groups [10,11,12].…”

Section: Introductionmentioning

confidence: 99%

“…The paper [6] contains results on limit automorphisms for inductive sequences of Toeplitz algebras which are closely related to the facts on the mappings of topological groups [10,11,12]. In [7,8] the authors introduce a topology associated to a partially ordered set and study its relation to properties of inductive limits arising from a system of C * -algebras over the partially ordered set which yields that topology. The existence of an isomorphism between the inductive limit of an inductive system of Toeplitz algebras over a directed set and the reduced semigroup C * -algebra for a semigroup in the group of rational numbers is shown in [7,9].…”

Section: Introductionmentioning

confidence: 99%

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Embedding Semigroup C*-algebras into Inductive Limits

Lipacheva

2019

Lobachevskii J Math

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The note is concerned with inductive systems of Toeplitz algebras and their * -homomorphisms over arbitrary partially ordered sets. The Toeplitz algebra is the reduced semigroup C *algebra for the additive semigroup of non-negative integers. It is known that every partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of some set. In our previous work we have studied a topology on this set of indexes. For every maximal upward directed subset we consider the inductive system of Toeplitz algebras that is defined by a given inductive system over an arbitrary partially ordered set and its inductive limit. Then for a base neighbourhood U a of the topology on the set of indexes we construct the C * -algebra B a which is the direct product of those inductive limits. In this note we continue studying the connection between the properties of the topology on the set of indexes and properties of inductive limits for systems consisting of C * -algebras B a and their * -homorphisms. It is proved that there exists an embedding of the reduced semigroup C * -algebra for a semigroup in the additive group of all rational numbers into the inductive limit for the system of C * -algebras B a .

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Section: Preliminariesmentioning

confidence: 99%

“…We consider the topology on the index set I which was introduced in [7,8]. For the convenience of the reader we recall the definition of this topology and its properties.…”

Section: Topology On An Index Setmentioning

confidence: 99%

“…Examples of different topological spaces (I, τ ) are contained in [7,8]. In particular, (I, τ ) may be: a non-Hausdorff space [ Here we give an example of a compact space.…”

Section: Topology On An Index Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Embedding Semigroup C*-algebras into Inductive Limits

Lipacheva

2019

Lobachevskii J Math

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On a Topology and Limits for Inductive Systems of C∗-Algebras over Partially Ordered Sets

2019

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Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of C * -algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of C * -algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those inductive limits, we construct different types of C * -algebras. In particular, for neighbourhoods of the topology on the set of indices we deal with the C * -algebras which are the direct products of those inductive limits. The present paper is concerned with the above-mentioned topology and the algebras arising from an inductive system of C * -algebras over a partially ordered set. We show that there exists a connection between properties of that topology and those C * -algebras.

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Extensions of Semigroups and Morphisms of Semigroup $ C^{*} $-Algebras

Lipacheva

2021

Sib Math J

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On Inductive Limits for Systems of C*-Algebras

Cited by 13 publications

References 13 publications

Embedding Semigroup C*-algebras into Inductive Limits

Embedding Semigroup C*-algebras into Inductive Limits

On a Topology and Limits for Inductive Systems of C∗-Algebras over Partially Ordered Sets

Extensions of Semigroups and Morphisms of Semigroup $ C^{*} $-Algebras

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