On The Extension of the Toeplitz Algebra by Isometries

TL;DR

This paper investigates the extensions of the Toeplitz algebra by isometries, revealing that irreducible actions lead to trivial extensions, while reducible cases can produce non-trivial extensions, advancing understanding of algebraic structures related to semigroup actions.

Contribution

It introduces the concept of cpie-extension of c6e-semigroups and characterizes when Toeplitz algebra extensions are trivial or non-trivial.

Findings

Irreducible actions yield only trivial extensions.

Non-trivial extensions exist in reducible representations.

The study clarifies conditions for extension triviality in Toeplitz algebras.

Abstract

We introduce the notion of \pi-extension of the semigroup \mathbb{Z}_+ and study the extensions of the Toeplitz algebras by isometric operators. We show that when the action of the Toeplitz algebra is irreducible all such extensions generate the same algebra, i.e. there are no non-trivial extensions of the Toeplitz algebra. Also we provide the examples of the non-trivial extensions of the Toeplitz algebra in case its representation is reducible.