Complete positivity of Rieffel’s deformation quantization by actions of \(\mathbb R^d\). (English) Zbl 1172.53055

Kaschek, Daniel; Neumaier, Nikolai; Waldmann, Stefan

J. Noncommut. Geom. 3, No. 3, 361-375 (2009).

The authors consider Rieffel’ s deformation by actions of \(\mathbb R^d\) in general and prove that every state of the underformed algebra can be deformed into a continuous field of states for the field of deformed algebras. Moreover, the authors give an explicit construction including a detailed study of the asymptotics of the deformed states for \(\hbar \rightarrow 0\). It turns out that the asymptotic expansion coincides in a precise sense with the formal deformations obtained in [H. Bursztyn and S. Waldmann, On positive deformations of \(\ast\)-algebras. Conférence Moshé Flato 1999: Quantization, deformations, and symmetries, Dijon, France, September 5–8, 1999. Volume II. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 22, 69–80 (2000; Zbl 0979.53098)].

Reviewer: Benjamin Cahen (Metz)

MSC:

53D55	Deformation quantization, star products
46L87	Noncommutative differential geometry
81R60	Noncommutative geometry in quantum theory
46L65	Quantizations, deformations for selfadjoint operator algebras

Keywords:

Rieffel deformation quantization; completely positive deformation; continuous fields

Citations:

Zbl 0979.53098

References:

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[2]	S. Beiser, H. Römer, and S. Waldmann, Convergence of the Wick star product. Comm. Math. Phys. 272 (2007), 25-52. · Zbl 1203.53089