Partial-isometric crossed products of dynamical systems by left LCM semigroups

TL;DR

This paper investigates a class of semigroup crossed product C*-algebras where actions are implemented by partial isometries, providing a structural understanding of Nica-Toeplitz algebras for product systems of Hilbert bimodules.

Contribution

It introduces a model for Nica-Toeplitz algebras associated with semigroup dynamical systems using partial isometries and establishes their behavior under exact sequences and tensor products.

Findings

Provides a structure theorem for the crossed product algebra

Shows the algebra models Nica-Toeplitz algebras of product systems

Demonstrates well-behaved properties under short exact sequences

Abstract

Let P be a left LCM semigroup, and $\alpha$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $\alpha$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.