Let $q = e^{i \theta } \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, that is, $u$ 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 $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$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 $q^{\prime}$-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 Lipschitz continuous function of $\theta $, as well as the result that the spectrum $\sigma (h_{\theta })$ is a $\frac{1}{2}$-Hölder continuous function in $\theta $ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial $p(u_{\theta },v_{\theta })$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. We define categorial Lévy processes on every tensor categoriy with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem.
We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).
We extend the Schoenberg correspondence for universal independences by Schürmann and Voß to the multifaced, multistate setting of Manzel and Schürmann, covering, e.g., Voiculescu's bifreeness as well as Bożejko and Speicher's c-free independence. At the same time, we free the proof in the single-face-single-state situation from its dependence on Muraki's classification theorem.
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