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Quantum maps and automorphisms. (English) Zbl 1077.53076
Marsden, Jerrold E. (ed.) et al., The breadth of symplectic and Poisson geometry. Festschrift in honor of Alan Weinstein. Boston, MA: Birkhäuser (ISBN 0-8176-3565-3/hbk). Progress in Mathematics 232, 623-654 (2005).
There are many different concepts of quantization of a symplectic map $$\chi$$ on a symplectic manifold $$(M,\omega)$$. For example, $$\chi$$ can be quantized as a Toeplitz quantum map (or Toeplitz Fourier integral operator), as an automorphism of the full observable algebra, or as an automorphism of the symbol algebra. In the paper, these concepts are discussed and compared in the case where $$(M,\omega)$$ is a Kähler manifold. The main result is a Toeplitz analogue of the Duistermaat-Singer theorem on automorphisms of the symbol algebra $$\Psi^*/\Psi^{-\infty}$$ of pseudodifferential operators [Commun. Pure Appl. Math. 29, 39–47 (1976; Zbl 0317.58017)] which states that, if $$H^1(S^*M,{\mathbb C})=0$$, then every order-preserving automorphism of $$\Psi^*/\Psi^{-\infty}$$ is either conjugation by an elliptic Fourier integral operator associated to the symplectic map or a transmission. In the paper, an analogous theorem is proven for Toeplitz operators and extended to the case where $$H^1(M,{\mathbb C})=0$$ is not assumed.
For the entire collection see [Zbl 1062.53002].

##### MSC:
 53D55 Deformation quantization, star products 53D50 Geometric quantization 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D05 Symplectic manifolds (general theory) 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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