Bergman空间中模轨道的一个可计算游荡且迹型向量
A computable wandering and tracelike vector for modular orbits in the Bergman space
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中文总结 AI 辅助
本文构造了一个可有效计算的函数，其轨道构成权重12的Bergman空间的标准正交基，从而实现了PSL(2,Z)的游荡且迹型向量，解决了Jones遗留的构造问题，并为相关von Neumann代数提供了可计算实现。
中文摘要 AI 辅助
我们构造了一个函数$\Phi $，使得在$\mathrm{PSL}(2,\mathbb{Z})$的表示下，其轨道构成权重$\alpha =12$的Bergman空间的一组标准正交基。此外，我们通过提供一种有效程序，证明$\Phi $作为上半平面上的全纯函数是可有效计算的（在可计算分析的精确意义上）。这构造了$\mathrm{PSL}(2,\mathbb{Z})$的一个游荡且迹型向量，其抽象存在性由Vaughan Jones爵士在其最后一篇论文中证明，而相应的构造当时被留作问题。该函数使用正交化和模化方法构建，尽管其本身不是模函数，却显示出模的痕迹。它为$M_{12}(\Gamma )$与其交换子之间的抽象反同构提供了一个可计算的实现向量，该交换子按Rădulescu的意义由尖点形式Toeplitz算子生成，而Voiculescu的结果为$M_{12}(\Gamma )$提供了随机矩阵模型。
英文摘要
We construct a function $Φ$ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $α=12$. Moreover, we show that $Φ$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra $M_{12}(Γ)$ and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(Γ)$.
