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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.
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
A matrix convex set is a set of the form $mathcal{S} = cup_{ngeq 1}mathcal{S}_n$ (where each $mathcal{S}_n$ is a set of $d$-tuples of $n times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We st
Let $epsilon>0$ be a positive number. Is there a number $delta>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ |uv-vu|<dt, $$ there is a continuous path o
An E_0-semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert sp
We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property tur