Boundedness in Strict Deformation Quantization

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 $\mathrm{S}(V)$ 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 $V_{\beta }^{\mathrm{\prime }}$. 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.