The Nica-Toeplitz algebras of dynamical systems over abelian lattice-ordered groups as full corners

Abstract

Consider the pair $(G,P)$ consisting of an abelian lattice-ordered discrete group $G$ and its positive cone $P$. Let $\alpha$ be an action of $P$ by extendible endomorphisms of a $C^{\ast }$-algebra $A$. We show that the Nica-Toeplitz algebra ${\mathcal{T}}_{\text{cov}}(A{\times }_{\alpha }P)$ is a full corner of a group crossed product $\mathcal{B}{⋊}_{\beta }G$, where $\mathcal{B}$ is a subalgebra of ${\ell }^{\infty }(G,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is given by the shift on ${\ell }^{\infty }(G,A)$. By using this realization, we show that the ideal $\mathcal{I}$ of ${\mathcal{T}}_{\text{cov}}(A{\times }_{\alpha }P)$ for which the quotient algebra ${\mathcal{T}}_{\text{cov}}(A{\times }_{\alpha }P)/\mathcal{I}$ is the isometric crossed product $A{\times }_{\alpha }^{\text{iso}}P$ is also a full corner in an ideal $\mathcal{J}{⋊}_{\beta }G$ of $\mathcal{B}{⋊}_{\beta }G$.

to appear in Houston J. Math