Inductive Sequences of Toeplitz Algebras and Limit Automorphisms

Abstract

© 2020, PleiadesT Publishing,T Ltd. Abstract: The note is concerned with inductive sequences of Toeplitz algebras. The Toeplitz algebra is the C*-subalgebra in the algebra of all bounded linear operators. This subalgebra is generated by the right shift operator on the Hilbert space of all square summable complex-valued functions defined on the additive semigroup of non-negative integers. We study the inductive sequences of Toeplitz algebras whose bonding *-homomorphisms are defined by arbitrary sequences of natural numbers. The inductive limits of such sequences are the reduced semigroup C*-algebras generated by representations for semigroups of non-negative rational numbers. We consider the limit *-endomorphisms of these inductive limits. Such an endomorphism is induced by a morphism between two copies of the same inductive sequence of Toeplitz algebras. We give the necessary and sufficient conditions for these endomorphisms to be *-automorphisms of C*-algebras. These criteria are formulated in algebraic, number-theoretical and functional terms.