We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia (J. Funct. Anal. 269:781-814, 2015) in the case $p = 2$, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant $C^\ast$ subalgebra of the bounded uniformly continuous functions as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols.

中文翻译：

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p-Fock 空间对应理论在 Toeplitz 代数中的应用

我们使用平移不变符号和运算符空间的对应理论证明了关于 $p$-Fock 空间上的 Toeplitz 代数理论的几个结果。最显着的结果是： 完整的 Toeplitz 代数是所有具有有界一致连续符号的 Toeplitz 算子的范数闭包。这概括了 J. Xia (J. Funct. Anal. 269:781-814, 2015) 在 $p = 2$ 的情况下获得的结果，该结果通过不同的方法得到了证明。此外，我们证明了每个具有有界一致连续函数的平移不变 $C^\ast$ 子代数作为其符号集的 Toeplitz 代数是由具有相同符号空间的 Toeplitz 算子线性生成的。