ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 2, pp. 7–18.
Allerton Press, Inc., 2018.
Original Russian Text
S.A. Grigoryan, A.Yu. Kuznetsova, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 10–22.
-Algebras Generated by Mappings.
Criterion of Irreducibility
S. A. Grigoryan
and A. Yu. Kuznetsova
Kazan State Power Engineering University
ul. Krasnosel’skaya 51, Kazan, 420066 Russia
Kazan Federal University
ul. Kremlyovskaya 18, Kazan, 420008 Russia
Received October 23, 2016
Abstract—We study the operator algebra associated with a self-mapping ϕ on a countable set X
which can be represented as a directed graph. The algebra is generated by the family of partial
isometries acting on the corresponding l
(X). We study the structure of involutive semigroup
multiplicatively generated by the family of partial isometries. We formulate the criterion when the
algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras.
In particular, we give the examples of nonisomorphic C
-algebras, which are the extensions by
compact operators of the algebra of continuous functions on the unit circle.
-algebra, partial isometry, positive operator, projection, compact operator,
Toeplitz algebra, extension of C
-algebra by compact operators.
In this paper we study an operator algebra associated with a self-mapping of a countable set such
that the preimage of each point is ﬁnite.
An algebraic approach to the theory of abstract dynamics systems was proposed by von Neumann
. The main idea of the algebraic theory of dynamical systems is in the construction of a C
-, as of von Neumann) that reﬂects the structure of the given dynamics system. For reversible
systems, the dynamics is given by a group of automorphisms on measurable spaces. This theory is well
investigated and enjoys wide acceptance [2–6].
The irreversible dynamics systems were investigated, e.g., in [7–11]. In  they consider a trans-
form T preserving the measure (in , type of measure) of the Lebesgue space. In [9, 10] they consider a
surjective mapping T : X −→ X,whereX is a compactum.
In [12, 13], we proposed the construction of a C
(X) generated by a self-mapping
ϕ : X −→ X of a set. Unlike in the cited articles, we assume the set X to be countable and have
no additional structure, but we do not assume the mapping ϕ to be reversible, we only assume it
to satisfy the condition card ϕ
(x) < ∞ for any x ∈ X. A family U of partial isometries acting on
(X), ﬁnite or countable, is connected with a pair (X, ϕ). These are the isometries that generate
(X) and, in addition, satisfy the relations: The sum of the initial and ﬁnal projections results into
noncommuting projections deﬁned by the mapping ϕ. Therefore, in some sense, C
(X) can be classiﬁed
as an algebra generated by partial isometries with some algebraic relations. The classical examples of