Dilations of $q$-Commuting Unitaries
Dilations of $q$-Commuting Unitaries
复制标题
$q$-通勤酉的扩张
DOI：
10.1093/imrn/rnaa093
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发表时间：
2022
影响因子：
1
通讯作者：
O. M. Shalit
中科院分区：
文献类型：
--
作者：
M. Gerhold;O. M. Shalit
Let(where), and letbe-commuting unitaries, that is,andare unitaries such that. In this paper, we find the optimal constantsuch thatcan be dilated to a pair of operators, whereandare commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$whereare the universal-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating-commuting unitaries to scalar multiples of-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”, we recover the fact that the normis a Lipschitz continuous function of, as well as the result that the spectrumis a-Hölder continuous function inwith respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
Let(where), and letbe-commuting unitaries, that is,andare unitaries such that. In this paper, we find the optimal constantsuch thatcan be dilated to a pair of operators, whereandare commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$whereare the universal-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating-commuting unitaries to scalar multiples of-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”, we recover the fact that the normis a Lipschitz continuous function of, as well as the result that the spectrumis a-Hölder continuous function inwith respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
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DOI：
10.1051/jphys:0199000510170180300
发表时间：
1990
期刊：
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影响因子：
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期刊：
Annales Henri Poincaré
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