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Convergent star products for projective limits of Hilbert spaces. (English) Zbl 1394.46003
Let $$V$$ be a vector space endowed with a locally convex topology and a continuous bilinear form. In a previous article [J. Geom. Phys. 81, 10–46 (2014; Zbl 1287.53073)], the second named author constructed a topology on the symmetric algebra $$S^{\bullet}(V)$$ which makes the star product that comes from the bilinear form continuous. Many aspects of this topology were studied; in particular, the construction is functorial. In the present paper, the authors consider the case that the topology on $$V$$ is the projective limit topology of Hilbert spaces, thus given by a family of Hilbert seminorms on $$V$$. In [loc. cit.], on each fixed symmetric power $$S^k(V)$$, a topology coming from the projective tensor product topology on $$V^{\otimes k}$$ was used. In the present situation, there is another natural topology, namely, the topology induced by the extensions of the Hilbert seminorms on $$S^k(V)$$. The latter leads to a different topology on $$S^{\bullet}(V)$$, which is studied in detail in this article. The authors show that, in the special case that $$V$$ is nuclear, the previous and the present construction of the topology coincide. In the special case that $$V$$ is a Hilbert space itself, the authors relate their results to earlier work of G. Dito [“Deformation quantization on a Hilbert space”, in: Proceedings of the COE International Workshop. 139–157 (2005; doi:10.1142/9789812775061_0009)].

##### MSC:
 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46L65 Quantizations, deformations for selfadjoint operator algebras
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