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)].


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


Zbl 0979.53098
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