Nets of graded $C^*$-algebras over partially ordered sets

Grigoryan, S. A.; Lipacheva, E. V.; Sitdikov, Аirat

doi:10.1090/spmj/1576

Cited by 12 publications

(4 citation statements)

References 13 publications

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“…The motivation for the present paper comes from algebraic quantum field theory [1]- [9] and our previous work on inductive systems of C * -algebras [10]- [12].…”

Section: Introductionmentioning

confidence: 99%

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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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“…The motivation for the present paper comes from algebraic quantum field theory [1]- [9] and our previous work on inductive systems of C * -algebras [10]- [12].…”

Section: Introductionmentioning

confidence: 99%

“…The paper [11] contains results on limit automorphisms for inductive sequences of Toeplitz algebras which are closely related to the facts on the mappings of topological groups [13], [14]. In [12] the authors deal with a net of C * -algebras associated to a net over a partially ordered set consisting of Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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“…The paper [15] deals with a net that is constructed by means of the semigroup C * -algebra generated by the path semigroup for a partially ordered. In [16] the authors consider a net of C * -algebras associated to a net over a partially ordered set consisting of Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

Inductive Limits for Systems of Toeplitz Algebras

Gumerov

2019

Lobachevskii J Math

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This article deals with inductive systems of Toeplitz algebras over arbitrary directed sets. For such a system the family of its connecting injective * -homomorphisms is defined by a set of natural numbers satisfying a factorization property. The motivation for the study of those inductive systems comes from our previous work on the inductive sequences of Toeplitz algebras defined by sequences of numbers and the limit automorphisms for the inductive limits of such sequences. We show that there exists an isomorphism in the category of unital C * -algebras and unital * -homomorphisms between the inductive limit of an inductive system of Toeplitz algebras over a directed set defined by a set of natural numbers and a reduced semigroup C * -algebra for a semigroup in the group of all rational numbers. The inductive systems of Toeplitz algebras over arbitrary partially ordered sets defined by sets of natural numbers are also studied.2010 Mathematics Subject Classification. 46L05, 47L40, 81T05 .

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“…The net constructed by means of the semigroup C * -algebra generated by the path semigroup for a partially ordered set is treated in [4]. In [5] the authors deal with a net consisting of C * -algebras associated to a net of Hilbert spaces over a partially ordered set.…”

Section: Introductionmentioning

confidence: 99%

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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Nets of graded $C^*$-algebras over partially ordered sets

Cited by 12 publications

References 13 publications

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

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

Inductive Limits for Systems of Toeplitz Algebras

Embedding Semigroup C*-algebras into Inductive Limits

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