This article is cited in 4 papers

Brief communications

$AF$-subalgebras of a $C^*$-algebra generated by a mapping

S. A. Grigoryan^a, A. Yu. Kuznetsova<sup>b

a</sup> Chair of Higher Mathematics, Kazan State Energy University, Kazan, Russia
^b Chair of General Relativity and Gravitation, Kazan State University, Kazan, Russia

Abstract: In this paper we consider a $ C^*$-subalgebra of the algebra of all bounded operators $B(l^2(X))$ on the Hilbert space $l^2(X)$ with one generating element $T_\varphi$ induced by a mapping $\varphi\colon X\to X$ of the set $X$ into itself. We prove that such a $C^*$-algebra has an $AF$-subalgebra and establish commutativity conditions for the latter. We prove that a $C^*$-algebra generated by a mapping produces a dynamic system such that the corresponding group of automorphisms is invariant on elements of the $AF$-subalgebra.

Keywords: $AF$-algebra, $C^*$-algebra, partial isometry.

UDC: 517.98

Received: 08.07.2009

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:3, 72–76

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