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 Izv. Vyssh. Uchebn. Zaved. Mat., 2018, Number 7, Pages 79–85 (Mi ivm9379)

Brief communications

On inductive limits for systems of $C^*$-algebras

R. N. Gumerova, E. V. Lipachevab, T. A. Grigoryanb

a Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia
b Kazan State Power Engineering University, 51 Krasnosel’skaya str., Kazan, 420066 Russia

Abstract: We consider a covariant functor from the category of an arbitrary partially ordered set into the category of $C^*$-algebras and their $*$-homomorphisms. In this case one has inductive systems of algebras over maximal directed subsets. The article deals with properties of inductive limits for those systems. In particular, for a functor whose values are Toeplitz algebras, we show that each such an inductive limit is isomorphic to a reduced semigroup $C^*$-algebra defined by a semigroup of rationals. We endow an index set for a family of maximal directed subsets with a topology and study its properties. We establish a connection between this topology and properties of inductive limits.

Keywords: covariant functor, direct product of $C^*$-algebras, inductive limit for an inductive system of $C^*$-algebras, partially ordered set, semigroup $C^*$-algebra, Toeplitz algebra, topology.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2018, 62:7, 68–73

Bibliographic databases:

Document Type: Article
UDC: 517.986

Citation: R. N. Gumerov, E. V. Lipacheva, T. A. Grigoryan, “On inductive limits for systems of $C^*$-algebras”, Izv. Vyssh. Uchebn. Zaved. Mat., 2018, no. 7, 79–85; Russian Math. (Iz. VUZ), 62:7 (2018), 68–73

Citation in format AMSBIB
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