超越等连续的 $\mathbb{Z}^d$-交叉积的度量谱几何
Metric Spectral Geometry of $\mathbb{Z}^d$-Crossed Products beyond Equicontinuity
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中文总结 AI 辅助
本文在 $\mathbb{Z}^d$-交叉积上构造线性扭曲谱三元组，利用适应权重补偿 Dirac 算子增长，证明其 $\kappa$-度量性，并应用于量子环面交叉积的连续性。
中文摘要 AI 辅助
我们从谱三元组 $(\mathfrak{A},\mathcal{H},D)$ 和具有适应权重的作用 $\alpha$ 出发，在 $C^\ast$-交叉积 $\mathfrak{A}\rtimes_\alpha\mathbb{Z}^d$ 上构造线性扭曲谱三元组。该权重补偿了动力学下基础 Dirac 算子的增长，同时保留了 $\mathbb{Z}^d$ 方向上的典型微分结构。构造直接在原始交叉积上进行，不要求作用是等连续的。我们证明，只要基础谱三元组是度量的，所得到的扭曲谱三元组对每个 $\kappa>0$ 都是 $\kappa$-度量的：其关联的 Monge--Kantorovich 度量在交叉积的状态空间上度量化弱*拓扑。我们的框架特别适用于保持基础三元组 Lipschitz 半范数定义域的每个作用。在这种情况下，可以选择指数权重，使得诱导的度量几何恰好限制为基础代数上的原始度量几何。在等连续情形下，权重可以选择为常数，扭曲消失，恢复标准的交叉积谱三元组。我们还建立了谱邻近性中收敛的一般有限模准则。作为应用，我们证明了与量子 $2$-环面被 $\operatorname{SL}(2,\mathbb{Z})$ 中固定自同构（包括 Anosov 自同构）交叉积相关的扭曲谱三元组，关于形变参数和平面度量的适当波动具有联合连续性。
英文摘要
We construct linearly twisted spectral triples on $C^\ast$-crossed products $\mathfrak{A}\rtimes_α\mathbb{Z}^d$, starting from a spectral triple $(\mathfrak{A},\mathcal{H},D)$ and an action $α$ admitting an adapted weight. The weight compensates for the growth of the base Dirac operator under the dynamics while preserving the canonical differential structure in the $\mathbb{Z}^d$-directions. The construction takes place directly on the original crossed product and does not require the action to be equicontinuous. We prove that, whenever the base spectral triple is metric, the resulting twisted spectral triple is $κ$-metric for every $κ>0$: its associated Monge--Kantorovich metric metrizes the weak* topology on the state space of the crossed product. Our framework applies, in particular, to every action preserving the domain of the Lipschitz seminorm of the base triple. In this case, an exponential weight may be chosen so that the induced metric geometry restricts exactly to the original metric geometry on the base algebra. In the equicontinuous case, the weight may be chosen constant, the twist disappears, and the standard crossed-product spectral triple is recovered. We also establish a general finite-mode criterion for convergence in the spectral propinquity. As an application, we prove joint continuity, with respect to the deformation parameter and suitable fluctuations of the flat metric, for the twisted spectral triples associated with crossed products of quantum $2$-tori by fixed automorphisms in $\operatorname{SL}(2,\mathbb{Z})$, including Anosov automorphisms.
发表机构
- University of Denver（丹佛大学）
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