Dilations of q-Commuting Unitaries

Abstract

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.

Dilations of unitary tuples

“…This result is originally due to Kesten [17,Theorem 3], who gives a probabilistic proof; in his notation, h f = 2dM (G, A, P ) where G is the free group F d , A the canonical set of generators, P assigns probability 1 2d to each generator, and M is the (infinite) matrix of transition propabilities of the associated symmetric random walk on G. See also [1, Theorem IV J] for a C*-algebraic proof (h f is a free operator in the sense [1, Definition III B] by [1, Theorem III E]) and [19] for a proof using free probability (the norm can be calculated from the general formula for the norm of free operators as in [1]). In this case, we can argue similar to [14,Theorem 6.1]. Let u = u f denote a d-tuple of free Haar unitaries.…”

“…Thus, we seek the convex combination t k x Θ,k of minimal norm. In [14], the case d = 2 was studied. With u θ denoting the universal pair satisfying u θ,2 u θ,1 = e iθ u θ,1 u θ,2 , the main result (Theorem 6.3) in that paper can be stated as follows:…”

“…Problem 1.1 and similar problems have come up in the setting of relaxation of spectrahedral inclusion problems [16], in interpolation problems for completely positive maps and the study of the structure of operator systems [9,10], and fit in the general paradigm of studying operator theory through dilations [29]. More recently, such problems have turned out to be connected to quantum information theory [6,7] as well as other aspects of mathematical physics and C*-algebras [14].…”

Dilations of unitary tuples

“…The above corollary is powerful enough to prove quite easily that for every selfadjoint ‐polynomial, the spectrum is a ‐Hölder continuous map, and from this one easily obtains the known results that the higher dimensional rotation algebras form a continuous field. We will not elaborate here on this method (see [15, Section 4] for details in the case ), since much stronger conclusions can be drawn by invoking an ingenious technique from [16].…”

“…Problem 1.1 and similar problems have come up in the setting of relaxation of spectrahedral inclusion problems [17], in interpolation problems for completely positive maps and the study of the structure of operator systems [10, 11], and fit in the general paradigm of studying operator theory through dilations [30]. More recently, such problems have turned out to be connected to quantum information theory [6, 7] as well as other aspects of mathematical physics and C*‐algebras [15].…”