C*-algebra generated by the paths semigroup

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

doi:10.1134/s1995080216060135

Cited by 9 publications

(11 citation statements)

References 11 publications

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“…Proof. In [13], it is shown that if K is an upward directed set then for each loop p one has the equivalence p ∼ i a for some a ∈ K. This means that every cycle χ p is trivial. Applying Theorem 3.2, we obtain the assertion of the theorem.…”

Section: Lemma 32 the Following Assertions Holdmentioning

confidence: 99%

“…For a ∈ K we denote by G a the set of all equivalence classes of loops whose base point is a. In [13] it is shown G a is a subgroup in S with the unit [i a ] and other properties of G a and S are described. In particular, it is proved that if K is an upward directed set then G a is trivial.…”

Section: Paths and Loops On A Partially Ordered Setmentioning

confidence: 99%

“…The set K is said to be path connected provided that for every a, b ∈ K there exists a path p such that ∂ 0 p = a, ∂ 1 p = b. It is shown in [13] that for a path-connected set K the isomorphism G a ∼ = G b holds whenever a, b ∈ K. In particular, the mapping σ ba : G a → G b defined by the formula…”

Section: Paths and Loops On A Partially Ordered Setmentioning

confidence: 99%

“…We have studied earlier the C * -algebras generated by representations of ordered semigroups [7,8,9,10,11,12]. The present paper is a continuation of the study begun in the article [13]. There we dealt with the C * -algebra generated by the path semigroup in a partially ordered set.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Grigoryan

Lipacheva

Sitdikov

2019

St. Petersburg Math. J.

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The paper deals with C * -algebras generated by a net of Hilbert spaces over a partially ordered set. The family of those algebras constitutes a net of C * -algebras over the same set. It is shown that every such an algebra is graded by the first homotopy group of the partially ordered set. We consider inductive systems of C * -algebras and their limits over maximal directed subsets. We also study properties of morphisms for nets of Hilbert spaces as well as nets of C * -algebras.

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Section: Lemma 32 the Following Assertions Holdmentioning

confidence: 99%

Section: Paths and Loops On A Partially Ordered Setmentioning

confidence: 99%

Section: Paths and Loops On A Partially Ordered Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Grigoryan

Lipacheva

Sitdikov

2019

St. Petersburg Math. J.

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“…In [1,2,3] the authors study nets containing C * -algebras of quantum observables for the case of curved spacetimes. 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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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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C*-algebra generated by the paths semigroup

Cited by 9 publications

References 11 publications

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

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

Embedding Semigroup C*-algebras into Inductive Limits

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

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