This article is cited in 11 papers

Brief communications

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

R. N. Gumerov^a, E. V. Lipacheva^b, T. A. Grigoryan<sup>b

a</sup> 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.

UDC: 517.986

Received: 19.03.2018

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

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