由满足扭曲交换关系的等距算子生成的$C^*$-代数的酉不变量
A unitary invariant for $C^*$-algebras generated by isometries with a twisted commutation relation
- Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究满足$V_1V_2=qV_2V_1$（$q$为无理旋转）的等距算子对生成的$C^*$-代数的酉分类，通过函数模型和半Fredholm指标构造酉不变量，扩展了Berger-Coburn-Lebow定理至非交换情形。
AI中文摘要：
我们研究由满足$V_1V_2=qV_2V_1$的等距算子对$(V_1, V_2)$生成的$C^*$-代数在酉等价意义下的分类问题，其中$q=e^{2\pi i\theta}$且$\theta$为无理数。利用一个函数模型，该模型在向量值Hardy空间上通过唯一关联的三元组$(\mathcal{F}, P, U)$（由Hilbert空间$\mathcal{F}$、正交投影$P$和酉算子$U$组成）实现纯$q$-交换对，我们通过其极小$q$-交换酉扩张生成的$C^*$-代数关于缺陷理想$\mathcal{I}_{\mathcal{D}} =\langle [V_1^*, V_1],[V_2^*, V_2]\rangle$的扩张来分析$C^*(V_1,V_2)$的内部结构。对于本质双重$q$-交换的对，我们证明$PUP|_{\operatorname{ran} P}$是半Fredholm算子，并证明$|\operatorname{ind} PUP|_{\operatorname{ran} P}|$在相应$C^*$-代数的酉等价下保持不变，从而为区分非等价对提供了明确判据。这项工作扩展了Berger、Coburn和Lebow于1978年关于交换等距算子的开创性结果，同时处理了该非交换设置中的关键技术细节：最显著的是，由于非交换性导致Gelfand理论的失效，以及用非交换环面关于紧算子理想的扩张取代经典的Toeplitz扩张。
英文摘要：
We investigate the classification of $C^*$-algebras generated by pairs of isometries $(V_1, V_2)$ satisfying $V_1V_2=qV_2V_1$, where $q=e^{2πiθ}$ with $θ$ irrational, up to unitary equivalence. Using a functional model that realizes pure $q$-commuting pairs on vector-valued Hardy spaces in terms of a uniquely associated triple $(\mathcal{F}, P, U)$ consisting of Hilbert space $\mathcal{F}$, an orthogonal projection $P$ and a unitary operator $U$, we analyze the internal structure of $C^*(V_1,V_2)$ via an extension of the $C^*$-algebra generated by its minimal $q$-commuting unitary extension by the defect ideal $\mathcal{I}_{\mathcal{D}} =\langle [V_1^*, V_1],[V_2^*, V_2]\rangle$. For pairs that are essentially doubly $q$-commuting, we show that $PUP|_{\operatorname{ran} P}$ is semi-Fredholm and prove that $|\operatorname{ind} PUP|_{\operatorname{ran} P}|$ is invariant under unitary equivalence of the corresponding $C^*$-algebras, thereby providing an explicit criterion for distinguishing non-equivalent pairs. This work extends the seminal 1978 results of Berger, Coburn, and Lebow for commuting isometries, while addressing the key technical nuances of this noncommutative setting: most notably, navigating the failure of Gelfand theory due to noncommutativity and replacing the classical Toeplitz extension by an extension of the noncommutative torus by the ideal of compact operators.
