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

Abstract

Suppose ${\Gamma }^{+}$ is the positive cone of a totally ordered abelian group $\Gamma$, and $(A,{\Gamma }^{+},\alpha )$ is a system consisting of a $C^{\ast }$-algebra $A$, an action $\alpha$ of ${\Gamma }^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_α^{\piso}Γ^{+}$ is a full corner in the subalgebra of $Ł({\ell }^2({\Gamma }^{+},A))$, and that if $\alpha$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{Γ^{+}}\otimes A)\times^{\iso}Γ^{+}$, 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 $A\times_α^{\piso}Γ^{+}$ such that the quotient is the isometric crossed product $A\times_α^{\iso}Γ^{+}$.

The paper is going to appear in J. Austral. Math. Soc

References in corpus (1)

• Partial-isometric crossed products by semigroups of endomorphisms

Cited by in corpus (2)

• The partial-isometric crossed products by semigroups of endomorphisms are Morita equivalent to crossed products by groups
• The Nica-Toeplitz algebras of dynamical systems over abelian lattice-ordered groups as full corners