Partial-isometric crossed products by semigroups of endomorphisms

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 $C^{\ast }$-algebra $A$. We consider a new kind of crossed-product $C^{\ast }$-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.

26 pages

Cited by in corpus (12)

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• Crossed products by endomorphisms of $C_0(X)$-algebras
• Crossed product of a C*-algebra by a semigroup of endomorphisms generated by partial isometries
• Exel's crossed product and crossed products by completely positive maps
• Iterating the Pimsner construction
• The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners
• The partial-isometric crossed products by semigroups of endomorphisms are Morita equivalent to crossed products by groups
• Exel and Stacey crossed products, and Cuntz-Pimsner algebras
• C*-algebras of commuting endomorphisms
• The Nica-Toeplitz algebras of dynamical systems over abelian lattice-ordered groups as full corners
• C*-algebras generalizing both relative Cuntz-Pimsner and Doplicher-Roberts algebras
• Stacey crossed products associated to Exel systems