The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners

TL;DR

This paper demonstrates that partial-isometric crossed products by semigroups of endomorphisms can be realized as full corners in certain operator algebras, linking them to isometric and group crossed products.

Contribution

It establishes that these crossed products are full corners in operator algebras and identifies their ideal structure, extending the understanding of their algebraic and analytical properties.

Findings

Partial-isometric crossed products are full corners in subalgebras of bounded operators.

When endomorphisms are automorphisms, the crossed product is a full corner of a group crossed product.

The ideal structure of the crossed product is characterized, relating it to isometric crossed products.

Abstract

Suppose $\Gamma^{+}$ is the positive cone of a totally ordered abelian group $\Gamma$, and $(A,\Gamma^{+},\alpha)$ is a system consisting of a $C^*$-algebra $A$, an action $\alpha$ of $\Gamma^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_{\alpha}^{\piso}\Gamma^{+}$ is a full corner in the subalgebra of $\L(\ell^{2}(\Gamma^{+},A))$, and that if $\alpha$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{\Gamma^{+}}\otimes A)\times^{\iso}\Gamma^{+}$, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of $A\times_{\alpha}^{\piso}\Gamma^{+}$ such that the quotient is the isometric crossed product $A\times_{\alpha}^{\iso}\Gamma^{+}$.