Partial-isometric crossed products by semigroups of endomorphisms

TL;DR

This paper introduces a new class of crossed-product $C^*$-algebras generated by endomorphisms of a $C^*$-algebra, characterized by partial isometries, and explores their structure and applications to Toeplitz algebras and product systems.

Contribution

It defines and analyzes a novel type of crossed product by semigroups of endomorphisms, linking it to Toeplitz algebras and providing detailed structural theorems.

Findings

Provides detailed structure theorems for actions by forward and backward shifts.

Establishes the connection between the new crossed products and Toeplitz algebras for product systems.

Demonstrates the richness and tractability of these crossed products in operator algebra theory.

Abstract

Let $\Gamma^+$ be the positive cone in a totally ordered abelian group $\Gamma$, and let $\alpha$ be an action of $\Gamma^+$ by endomorphisms of a $C^*$-algebra $A$. We consider a new kind of crossed-product $C^*$-algebra $A\times_\alpha\Gamma^+$, which is generated by a faithful copy of $A$ and a representation of $\Gamma^+$ as partial isometries. We claim that these crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules, as recently studied by Fowler, and we illustrate this by proving detailed structure theorems for actions by forward and backward shifts.