C^*-envelopes of tensor algebras for multivariable dynamics

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.2000 Mathematics Subject Classification. 47L55, 47L40, 46L05, 37B20, 37B99.

Dilating covariant representations of the non-commutative disc algebras

“…This can be used to find explicit information about the C * -envelope. See Peters [27] for the one variable case and [13] for the C * -envelope a multivariable dynamical system.…”

Operator Algebras and C*-correspondences: A Survey

“…This idea appears first in [14] for a different crossed product than the one presented here. We start with the pertinent definitions.…”

“…For e ∈ G (1) [14]). Let X ≡ {u, v} and consider the maps σ i : X → X , i = 1, 2, with σ i (u) = v and σ i (v) = v. Set σ ≡ (σ 1 , σ 2 ) and let O (X ,σ) be the Cuntz-Pimsner algebra associated with the multivariable system (X , σ), which by [13] is the C * -envelope of the associate tensor algebra.…”

“…We now follow the arguments of [14]. In order to obtain O (X ,σ) as a full corner of an injective Cuntz-Pimsner algebra, Davidson and Roydor add a tail to the multivariable system.…”

KMS states on Pimsner algebras associated with C⁎-dynamical systems

“…It goes back to the work of Davidson and Katsoulis on classical systems [12], exploited further by Davidson and Roydor [13], and by the author with Katsoulis [22,23]. It is not necessary for our purposes to review the whole theory of C*-correspondences.…”