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Register a new userFull Cuntz-Krieger dilations via non-commutative boundaries
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We apply Arvesons non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $mathcal T_+(G)$ is the Cuntz-Krieger algebra $mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating free products of operator algebras. These generalizations rely on results of independent interest on complete injectivity and a characterization of representations with the unique extension property for free products of operator algebras.
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Motivated by the theory of Cuntz-Krieger algebras we define and study $ C^ast $-algebras associated to directed quantum graphs. For classical graphs the $ C^ast $-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to $ KK $-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these $ C^ast $-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $ KK $-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
We construct a covariant functor from the topological torus bundles to the so-called Cuntz-Krieger algebras; the functor maps homeomorphic bundles into the stably isomorphic Cuntz-Krieger algebras. It is shown, that the K-theory of the Cuntz-Krieger algebra encodes torsion of the first homology group of the bundle. We illustrate the result by examples.
In this article, we present a new method to study relative Cuntz-Krieger algebras for higher-rank graphs. We only work with edges rather than paths of arbitrary degrees. We then use this method to simplify the existing results about relative Cuntz-Krieger algebras. We also give applications to study ideals and quotients of Toeplitz algebras.
Strengthening classical results by Bratteli and Kishimoto, we prove that two subshifts of finite type are shift equivalent in the sense of Williams if and only if their Cuntz-Krieger algebras are equivariantly stably isomorphic. This provides an equivalent formulation of Williams problem from symbolic dynamics in terms of Cuntz-Krieger C*-algebras. To establish our results, we apply works on shift equivalence and strong Morita equivalence of C*-correspondences due to Eleftherakis, Kakariadis and Katsoulis. Our main results then yield K-theory classification of C*-dynamical systems arising from Cuntz-Krieger algebras.
Let $q = e^{i theta} in mathbb{T}$ (where $theta in mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_theta$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that [ c_theta = frac{4}{|u_theta+u_theta^*+v_theta+v_theta^*|}, ] where $u_theta, v_theta$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q$-commuting unitaries. The techniques that we develop allow us to give new and simple dilation theoretic proofs of well known results regarding the continuity of the field of rotations algebras. In particular, for the so-called Almost Mathieu Operator $h_theta = u_theta+u_theta^*+v_theta+v_theta^*$, we recover the fact that the norm $|h_theta|$ is a Lipshitz continuous function of $theta$, as well as the result that the spectrum $sigma(h_theta)$ is a $frac{1}{2}$-Holder continuous function in $theta$ with respect to the Hausdorff metric. In fact, we obtain this Holder continuity of the spectrum for every selfadjoint $*$-polynomial $p(u_theta,v_theta)$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
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