We prove that to every connected Abelian subgroup H of the biholomorphisms of the unit ball ${\mathbb{B}}^n$ we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative $C^{⁎}$-algebra on every weighted Bergman space. These symbols are of the form $a(z)=f({\mu }^H(z))$, where ${\mu }^H$ is the moment map for the action of H on ${\mathbb{B}}^n$. We show that, for this construction, if H is a maximal Abelian subgroup, then the symbols introduced are precisely the H-invariant symbols. We provide the explicit computation of moment maps to obtain special sets of symbols described in terms of coordinates. In particular, it is proved that our symbol sets have as particular cases all symbol sets from the current literature that yield Toeplitz operators generating commutative $C^{⁎}$-algebras on all weighted Bergman spaces on the unit ball ${\mathbb{B}}^n$. Furthermore, we exhibit examples that show that some of the symbol sets introduced in this work have not been considered before. Finally, several explicit formulas for the corresponding spectra of the Toeplitz operators are presented. These include spectral integral expressions that simplify the known formulas for maximal Abelian subgroups for the unit ball.

我们证明了单位球的双同态的每个连接的Abelian子群H${\mathbb{乙}}^{ñ}$ 我们可以关联一组有界符号，其相应的Toeplitz运算符生成可交换的 $C^{⁎}$-每个加权Bergman空间上的-代数。这些符号的形式$\mathrm{一种}（ž）=F（{\mu }^H（ž））$， 在哪里 ${\mu }^H$是当下地图的作用^ h上${\mathbb{乙}}^{ñ}$。我们表明，对于这种构造，如果H是最大的Abelian子组，则引入的符号恰好是H不变符号。我们提供矩量图的显式计算，以获得以坐标形式描述的特殊符号集。特别地，证明了我们的符号集在特殊情况下具有当前文献中的所有符号集，这些符号集产生产生交换性的Toeplitz算符$C^{⁎}$-单位球上所有加权Bergman空间上的-代数 ${\mathbb{乙}}^{ñ}$。此外，我们展示了一些示例，这些示例表明在此工作中引入的一些符号集以前没有被考虑过。最后，给出了Toeplitz算子对应光谱的几个显式公式。这些包括频谱积分表达式，这些表达式简化了单位球的最大Abelian子组的已知公式。