Boundedness in Strict Deformation Quantization
Abstract
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 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.
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BibTeX
@misc{heins2026boundedness,
title = {{Boundedness in Strict Deformation Quantization}},
author = {Michael Heins},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16250},
eprint = {2609.16250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16250},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boundedness in Strict Deformation Quantization
AU - Michael Heins
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16250
UR - https://arxiv.org/abs/2609.16250
AB - 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 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.
ER -