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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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


Authors: S. A. Grigoryan, E. V. Lipacheva and A. S. Sitdikov
Translated by: the authors
Original publication: Algebra i Analiz, tom 30 (2018), nomer 6.
Journal: St. Petersburg Math. J. 30 (2019), 901-915
MSC (2010): Primary 47L30, 46M40; Secondary 47L90
DOI: https://doi.org/10.1090/spmj/1576
Published electronically: September 16, 2019
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Abstract: 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 algebra is graded by the first homotopy group of the partially ordered set. Inductive systems of $ C^*$-algebras and their limits over maximal directed subsets are considered, as well as properties of morphisms for nets of Hilbert spaces and nets of $ C^*$-algebras.


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Additional Information

S. A. Grigoryan
Affiliation: Kazan State Power Engineering University, 51 Krasnoselskaya st., 420066 Kazan, Russia
Email: gsuren@inbox.ru

E. V. Lipacheva
Affiliation: Kazan State Power Engineering University, 51 Krasnoselskaya st., 420066 Kazan, Russia
Email: elipacheva@gmail.com

A. S. Sitdikov
Affiliation: Kazan State Power Engineering University, 51 Krasnoselskaya st., 420066 Kazan, Russia
Email: airat_vm@rambler.ru

DOI: https://doi.org/10.1090/spmj/1576
Keywords: $C^*$-algebra, graded $C^*$-algebra, partially ordered set, net of $C^*$-algebras, net of Hilbert spaces, path semigroup, the first homotopy group, inductive limit
Received by editor(s): December 1, 2017
Published electronically: September 16, 2019
Article copyright: © Copyright 2019 American Mathematical Society