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

##### MSC:
 53D55 Deformation quantization, star products 46L87 Noncommutative differential geometry 81R60 Noncommutative geometry in quantum theory 46L65 Quantizations, deformations for selfadjoint operator algebras
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##### References:
 [1] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111 (1978), 61-110; Deformation theory and quantization. II. Physical applications. ibid. 111 (1978), 111-151. · Zbl 0377.53024 [2] S. Beiser, H. Römer, and S. Waldmann, Convergence of the Wick star product. Comm. Math. Phys. 272 (2007), 25-52. · Zbl 1203.53089
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