AI 中文总结
本文通过复解析方法解释严格形变量子化中张量代数上的拓扑，证明其可视为有界Fréchet全纯函数代数，并推广了整函数阶的概念，同时给出全纯性与完备性的新结果。
AI 中文摘要
本文对张量代数上的某些拓扑给出了复解析解释，这些拓扑是严格形变量子化中的核心工具。我们证明了，对于桶型核DF空间$V$的对称张量代数$\operatorname{S}(V)$，赋予这些拓扑中的任意一个，都可理解为强对偶空间$V'_\beta$上的有界Fréchet全纯函数代数。其中最粗的拓扑对应于在有界子集上的一致收敛拓扑。较细的拓扑则提供了整函数阶的无穷维推广。为得出这些结果，我们研究了局部凸空间之间全纯映射的有界性与连续性之间的相互作用。作为副产品，我们建立了Fréchet全纯性的一个简单充分判据，以及若干有界函数空间的完备性结果。
英文摘要
This paper provides a complex-analytic interpretation of certain topologies on the tensor algebra, which are a central tool within strict deformation quantization. We prove that the symmetric tensor algebra $\operatorname{S}(V)$ of a barelled nuclear DF-space $V$ endowed with any of these topologies may be understood as an algebra of bounded Fréchet holomorphic functions on the strong dual space $V'_β$. The coarsest of the topologies then corresponds to the topology of uniform convergence on bounded subsets. The finer ones provide an infinite-dimensional generalization of the order of an entire holomorphic function. To facilitate these results, we study the interplay between boundedness and continuity of holomorphic mappings between locally convex spaces. As a byproduct, we establish a simple sufficient criterion for Fréchet holomorphy as well as several completeness results for spaces of bounded functions.
Comments34 pages, based on Chapter 2 of the PhD Thesis arXiv:2504.12862
