[Paper Review] C*-envelopes of tensor algebras for multivariable dynamics

Kenneth R. Davidson, Jean Roydor|ArXiv.org|Nov 29, 2008

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper provides a concrete realization of the C*-envelope of the tensor algebra associated with a multivariable dynamical system $(X,\sigma)$, showing it is isomorphic to a crossed product by an endomorphism and a groupoid C*-algebra in the surjective case, and a full corner of such an algebra in the non-surjective case. The key result is that when $X$ is compact, the C*-envelope is simple if and only if the system is minimal.

ABSTRACT

We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal.

Motivation & Objective

• To provide a concrete, explicit description of the C*-envelope of the tensor algebra $\mathcal{A}(X,\sigma)$ associated with a multivariable dynamical system.
• To extend Peters' crossed product description of the C*-envelope in the $n=1$ case to the multivariable setting ($n \geq 2$), replacing automorphism actions with endomorphism actions.
• To characterize the simplicity of the C*-envelope in the compact case via a dynamical condition on the system.
• To establish a connection between the C*-envelope and groupoid C*-algebras, and to relate ideal theory in the C*-envelope to invariant sets in the associated dynamical system.

Proposed method

• Construct the C*-envelope as a crossed product $\mathfrak{B} \times_\alpha \mathbb{N}$ by an endomorphism $\alpha$, where $\mathfrak{B}$ is an inductive limit of homogeneous C*-algebras over the space $\tilde{X}$, the inverse limit of the system.
• Define the endomorphism $\alpha$ on $\mathfrak{B}$ via the shift action on the semigroup $\mathbb{F}_n^+$, and show that $\mathrm{C}^*_{\text{e}}(\mathcal{A}(X,\sigma)) \cong \mathfrak{B} \times_\alpha \mathbb{N}$ in the surjective case.
• Represent the C*-envelope as a groupoid C*-algebra using the transformation groupoid associated with the endomorphism action on $\tilde{X}$.
• For non-surjective systems, show that the C*-envelope is a full corner of the C*-envelope of the surjective system obtained via the surjective hull construction.
• Use the theory of maximal dilations and boundary representations to construct the C*-envelope explicitly, avoiding irreducibility constraints.
• Apply Paschke's theorem on simplicity of crossed products by endomorphisms to characterize simplicity of the C*-envelope in terms of the absence of nontrivial $\alpha$-invariant ideals in $\mathfrak{B}$.

Experimental results

Research questions

• RQ1 How can the C*-envelope of the tensor algebra $\mathcal{A}(X,\sigma)$ for a multivariable dynamical system be described explicitly in terms of operator algebras?
• RQ2 Is the C*-envelope of $\mathcal{A}(X,\sigma)$ isomorphic to a crossed product by an endomorphism when $n \geq 2$, and if so, what is the structure of this crossed product?
• RQ3 What is the relationship between the minimality of the system $(X,\sigma)$ and the simplicity of its C*-envelope?
• RQ4 Can the C*-envelope be realized as a groupoid C*-algebra, and how does this realization reflect the dynamics of $\sigma$?
• RQ5 What dynamical condition on $X$ is equivalent to the simplicity of $\mathrm{C}^*_{\text{e}}(\mathcal{A}(X,\sigma))$ when $X$ is compact?

Key findings

• The C*-envelope of $\mathcal{A}(X,\sigma)$ is isomorphic to a crossed product $\mathfrak{B} \times_\alpha \mathbb{N}$ by an endomorphism $\alpha$ when the system is surjective, and to a full corner of such a crossed product when not surjective.
• The C*-envelope is isomorphic to a groupoid C*-algebra associated with the transformation groupoid of the endomorphism action on the inverse limit space $\tilde{X}$.
• When $X$ is compact, the C*-envelope is simple if and only if the system $(X,\sigma)$ is minimal.
• Minimality of $X$ is equivalent to the absence of proper $\alpha$-bi-invariant ideals in $\mathfrak{B}$, and this condition ensures simplicity of the crossed product $\mathfrak{B} \times_\alpha \mathbb{N}$ via Paschke's theorem.
• For $n=1$, the C*-envelope of the disc algebra is $C(\mathbb{T})$, which is not simple, showing that the $n \geq 2$ condition is essential for the simplicity criterion to hold.
• The construction shows that the C*-envelope is Morita equivalent to the C*-envelope of the surjective hull system, and simplicity is preserved under this equivalence.