Limits of Inductive Sequences of Toeplitz--Cuntz Algebras

S. A. Grigoryan ; R. N. Gumerov ; E. V. Lipacheva

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematics of Quantum Technologies, Tome 313 (2021), pp. 67-77.

Résumé

We consider inductive sequences of Toeplitz–Cuntz algebras. The connecting homomorphisms of such a sequence are defined by a finite set of sequences of positive integers. We prove that the inductive limit of such a sequence of Toeplitz–Cuntz algebras is isomorphic to the reduced semigroup $C^*$-algebra constructed for the unitalization of the free product of a finite family of semigroups of positive rational numbers. We show that the limit of the inductive sequence of Toeplitz–Cuntz algebras defined by a finite set of sequences of positive integers is a simple $C^*$-algebra.

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