AI 中文总结
本文为$\mathbb{T}^m$-不变Toeplitz算子代数建立局部化框架，通过符号延拓识别局部商代数，并显式描述极大理想空间与Gelfand变换。
AI 中文摘要
我们研究单位球上加权Bergman空间上的$\mathbb{T}^m$-不变Toeplitz算子。一个自然的酉变换将每个这样的算子表示为作用在多球上的Toeplitz算子限制的直和。在$L^\infty$-值角连续性假设下，相应的符号从$\mathbb{N}_0^m$扩展到由$k$-拟径向Toeplitz算子生成的$C^*$-代数的极大理想空间上的范数连续族。该扩展为Toeplitz算子代数提供了局部化框架。作为应用，我们考虑通过添加符号在混合酉作用和圆作用下不变的Toeplitz算子而得到的交换Banach代数。我们在有限坐标层和无穷远处识别其局部商代数，并利用这些识别来显式描述其极大理想空间和Gelfand变换。
英文摘要
We study $\mathbb T^m$-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an $L^\infty$-valued angular continuity assumption, the corresponding symbols extend from $\mathbb N_0^m$ to a norm-continuous family indexed by the maximal ideal space of the $C^*$-algebra generated by $k$-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.
