Operator algebras for multivariable dynamics

Abstract

Let X be a locally compact Hausdorff space with n proper continuous self maps σ i : X → X for 1 ≤ i ≤ n. To this we associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ ) and the semicrossed product C 0 (X) × τ F + n . We develop the necessary dilation theory for both models. In particular, we exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.We introduce a new concept of conjugacy for multidimensional systems, called piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from the algebraic structure of either A(X, σ) or C 0 (X) × σ F + n . Various classification results follow as a consequence. For example, if n = 2 or 3, or the space X has covering dimension at most 1, then the tensor algebras are algebraically isomorphic (or completely isometrically isomorphic) if and only if the systems are piecewise topologically conjugate.We define a generalized notion of wandering sets and recurrence. Using this, it is shown that A(X, σ) or C 0 (X) × σ F + n is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each σ i being surjective and v-recurrent points being dense for each v ∈ F + n .

Operator Algebras and C*-correspondences: A Survey

“…The tensor algebra T + X of a C * -correspondence A X A is the norm-closed subalgebra of T X generated by all elements of the form π ∞ (a), t n ∞ (ξ), a ∈ A,ξ ∈ X n , n ∈ N. The tensor algebras for C * -correspondences were pioneered by Muhly and Solel in [40]. They form a broad class of non-selfadjoint operator algebras which includes as special cases Peters' semicrossed products [45], Popescu's non-commutative disc algebras [47], the tensor algebras of graphs (introduced in [40] and further studied in [29]) and the tensor algebras for multivariable dynamics [13], to mention but a few. For more examples, see [31].…”

“…In the non-selfadjoint literature, these algebras are much more recent. In [13] Davidson and the second named author introduced the tensor algebra T (A,α) for a multivariable dynamical system (A, α). It turns out that T (A,α) is completely isometrically isomorphic to the tensor algebra for the C * -correspondence (X α , A, ϕ α ).…”

“…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.…”

Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

“…It fits into what seems to be an emerging line of research where Banach algebras of crossed product (or related) type are considered that are associated with (abstract) dynamical systems, but that are not C * -algebras or closed subalgebras of C * -algebras. We refer to [3,4,5,6,8,9,18,19,20] as examples of this development. These new algebras present an extra challenge compared to C * -algebras, because the latter with their rigidity properties are still reasonably manageable, and have a relatively uncomplicated-though still far from trivial-structure.…”

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

“…First of all we are inspired by the growing interest on the structure of the KMS states that involves: the Cuntz algebra [35], Cuntz-Krieger algebras [14], Hecke algebras [3], C*-algebras associated with subshifts [32], Pimsner algebras [26], the Toeplitz algebra of N ⋊ N × [27,28], C*-algebras of dilation matrices [29], C*-algebras of selfsimilar actions [30], topological dynamics [1,37,38], higher-rank graphs [2], and Nica-Pimsner algebras [21]. Secondly we are interested in analyzing further equivalence relations on multivariable classical systems [12,23]. Our motivation relies on the interaction of that theory with other fields of mathematics.…”