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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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