[Paper Review] Representations of C*-dynamical systems implemented by Cuntz families

Evgenios T. A. Kakariadis, Justin R. Peters|arXiv (Cornell University)|Dec 22, 2012

Advanced Operator Algebra Research35 references22 citations

TL;DR

This paper investigates non-selfadjoint operator algebras associated with C*-dynamical systems $(A,\alpha)$ via representations of Cuntz families, establishing that the C*-envelope of the algebra $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ is a full corner of the twisted crossed product $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, providing a non-automorphic analogue of Stacey's result for $n>1$. The work resolves a key gap by showing that the canonical $*$-epimorphism from Stacey's crossed product to the twisted crossed product is not injective when $n>1$ and $\alpha$ is not an automorphism.

ABSTRACT

Given a dynamical system $(A,\al)$ where $A$ is a unital $\ca$-algebra and $\al$ is a (possibly non-unital) *-endomorphism of $A$, we examine families $(π,\{T_i\})$ such that $π$ is a representation of $A$, $\{T_i\}$ is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of non-selfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest $\ca$-algebra they generate, namely the $\ca$-envelope. We then relate each occurrence of the $\ca$-envelope to (a full corner of) an appropriate twisted crossed product. We provide a counterexample to show the extent of this variety. In the context of $\ca$-algebras, these results can be interpreted as analogues of Stacey's famous result, for non-automorphic systems and $n>1$. Our study involves also the one variable generalized crossed products of Stacey and Exel. In particular, we refine a result that appears in the pioneering paper of Exel on (what is now known as) Exel systems.

Motivation & Objective

• To extend Stacey's multiplicity-$n$ crossed product result to non-automorphic $*$-endomorphisms when $n>1$, where the original isomorphism fails.
• To study the C*-envelope of non-selfadjoint operator algebras generated by representations of $A$ and Toeplitz-Cuntz families satisfying a covariance relation.
• To relate the C*-envelope to twisted crossed products via full corners, particularly $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, in the non-automorphic case.
• To clarify the distinction between the universal C*-algebra $A\times_{\alpha}^{n}\mathbb{N}$ and the C*-envelope of the non-selfadjoint algebra $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$, showing they are not isomorphic.

Proposed method

• Define two non-selfadjoint operator algebras: $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ and $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$, as universal algebras satisfying a covariance relation involving a Cuntz family $\{T_i\}$ and a representation $\pi$ of $A$.
• Use Arveson's program on the C*-envelope to identify the minimal C*-algebra generated by these non-selfadjoint algebras.
• Construct a canonical $*$-epimorphism from Stacey's universal C*-algebra $A\times_{\alpha}^{n}\mathbb{N}$ to the twisted crossed product $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, and show it is not injective via a counterexample.
• Demonstrate that the C*-envelope of $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ is a full corner of $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, generalizing Stacey's result to non-automorphic systems.
• Use the Fourier transform on the $\ell^1$-dense subalgebra $\ell^1(A,\alpha,\mathbb{F}_n^+)$ to show that the embedding into $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ is completely contractive but not completely isometric.
• Analyze the ideal structure and the Šilov ideal to confirm that the C*-envelope is minimal and isometrically reflects the non-selfadjoint algebra.

Experimental results

Research questions

• RQ1 Is the canonical $*$-epimorphism from Stacey's crossed product $A\times_{\alpha}^{n}\mathbb{N}$ to the twisted crossed product $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ injective when $\alpha$ is not an automorphism and $n>1$?
• RQ2 Can the C*-envelope of the non-selfadjoint algebra $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ be realized as a full corner of a twisted crossed product?
• RQ3 Does the C*-envelope of $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$ coincide with Stacey's universal C*-algebra $A\times_{\alpha}^{n}\mathbb{N}$?
• RQ4 Is the canonical embedding of $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$ into $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ completely isometric?
• RQ5 What is the relationship between the $\ell^1$-dense subalgebra $\ell^1(A,\alpha,\mathbb{F}_n^+)$ and the C*-envelope of $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$?

Key findings

• The C*-envelope of $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ is isomorphic to a full corner of the twisted crossed product $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, providing a non-automorphic generalization of Stacey's result for $n>1$.
• The canonical $*$-epimorphism $A\times_{\alpha}^{n}\mathbb{N} \to A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ is not injective when $n>1$ and $\alpha$ is not an automorphism, as shown by a counterexample using the non-vanishing of $S_1^*S_1S_2$.
• The C*-envelope of $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$ is isomorphic to $A\times_{\alpha}^{n}\mathbb{N}$, confirming that the universal C*-algebra is the minimal C*-algebra generated by the non-selfadjoint algebra.
• The canonical embedding $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+} \to A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ is completely contractive but not completely isometric, implying that the C*-envelope is strictly smaller than the image.
• The $\ell^1$-dense subalgebra $\ell^1(A,\alpha,\mathbb{F}_n^+)$ admits a Fourier transform and embeds injectively into $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$, but this embedding is not completely isometric.
• The C*-envelope of $A_{\text{nd}}\times_{\alpha}^{t}\mathcal{T}_{n}^{\!+}$ and $A_{\text{nd}}\times_{\alpha}\mathcal{T}_{n}^{\!+}$ are not isomorphic, as the former has a full corner in $A_{\infty}\rtimes_{\alpha_{\infty}}\mathcal{O}_{n}$ while the latter does not.