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Deformation quantization in infinite dimensional analysis. (English) Zbl 1172.53056
Blath, Jochen (ed.) et al., Trends in stochastic analysis. Papers dedicated to Professor Heinrich von Weizsäcker on the occasion of his 60th birthday. Cambridge: Cambridge University Press (ISBN 978-0-521-71821-9/pbk). London Mathematical Society Lecture Note Series 353, 303-325 (2009).
This paper reviews different aspects of deformation quantization. Let us recall that the simplest case of a star-product is the Moyal star-product on ${ℝ}^{n}$. Several works used this star-product in order to define a star-product on any symplectic manifold $M$ (for instance, the standard Fedosov quantization). With Kontsevich’s formality, the existence has been generalized to any Poisson manifold. In this work, the author replaces the algebra $C\left(M\right)$ by a Frechet algebra. The notion of equivalences of two star-products can be extended to this situation. Considering specific cases of functions, the author studies to types of deformations: continuous deformations and differential deformations. The algebra of functionals related to some infinite-dimensional manifolds and the quantization of the free path space of a manifold are finally discussed.
##### MSC:
 53D55 Deformation quantization, star products 60H07 Stochastic calculus of variations and the Malliavin calculus