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Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson brackets and associative algebras which deform in non-commutative direction algebra of functions on a smooth manifold. However most of these achievements have a algebraic flavor, my question concerns "adding analysis" to algebraic achievements.

Main belief: It is natural to believe that for a compact symplectic manifold $(M,\omega)$ one can construct a $C^*$-algebra which is "quantization" of the algebra of functions on $M$.

Focused question: Can one characterize such $C^*$-algebras ? (In "classical limit" we have a simple description "compact symplectic manifold " one may hope that there can be some simple description in "quantum" case).

Further questions: what is known in the direction of such belief ? What about existence and examples ? What about special properties of such algebras ?

"Smooth functions" in such algebra should be isomorphic to deformation quantization of smooth functions on $M$ for $h=1$ (respectively $h=1/k, k \in Z$ corresponds to $(M, k\omega)$).

The example of such situation is torus $T^2$ with standard symplectic form which "corresponds" to noncommutative torus $C^*$-algebra.

It is also natural to expect that for the case $[\omega] \in H^2(M,Z) $ (i.e. integrality condition) such $C^*$-algebra has a UNIQUE finite-dimensional irreducible representation which dimension is given by $ \int_{M} e^\omega Td(M)$ - index formula. (Which in the case of flat $M$ is just $ 1/n! $*(symplectic volume) the fact which was somehow known from early days of quantum theory (probably goes back to Zommerfeld or Heisenberg, at least it is mentioned in Landau-Lifshitz textbook as "number of quantum states corresponds to classical $dpdq / h$ bits of volume" ). In the case of quantum torus $AB= q BA$ it has n-dimensional irrep iff $q^n=1$, that indeed corresponds to the case quantization of torus with symplectic volume equal to $n$, i.e. $[\omega] \in H^2(M,Z) $. (See another example in MO231072, especially remark 3.

In the case $M$ is Kaehler one my expect that such irrep can be constructed by Berezin-Toeplitz quantization (see e.g. Schlichenmaier ).

The algebra should probably has a trace functional (i.e. $Tr:A\to C, Tr(AB)=Tr(BA)$) which should correspond to integration over $M$ (here we appeal to a kind of deformation quantization picture - that the algebra as a linear space is the same as algebra of functions on $M$, but the product is deformed to non-commutative).

One may push further the classical-to-quantum dictionary: "states" in quantum and classical cases might be related, Lagrangian submanifolds should correspond to certain one-sided ideals in the algebra, so on...

There are lots of compact symplectic manifolds for example complex projective spaces and their submanifolds - can one describe corresponding $C^*$-algebras ?

The simplest example of $S^2$ is discussed in MO231072. One can also discuss flag manifolds in the same vein, and also complex projective spaces and more generally toric manifolds in different, but explicit manner.

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  • $\begingroup$ I think this can be done using geometric quantization. $\endgroup$ – Qiaochu Yuan Feb 9 '16 at 20:25
  • $\begingroup$ Are you interested in what kinds of quantization? It looks like you're looking for strict deformation quantization. If this is the case, geometric quantization of symplectic groupoids gives the answer since you're just dealing with symplectic manifolds (which are, then, integrable). $\endgroup$ – user40276 Feb 9 '16 at 23:39
  • $\begingroup$ About the C^* algebras you would have to classify all groupoid convolution algebras that arises from the fundamental groupoid or the pair groupoid. This should not be very difficult. $\endgroup$ – user40276 Feb 9 '16 at 23:45
  • $\begingroup$ @QiaochuYuan I am not sure ... also there can be different viewpoints on geometric quantization. $\endgroup$ – Alexander Chervov Feb 13 '16 at 19:36
  • $\begingroup$ @user40276 quantization is unique up to certain things. Deformation quantization is very good, but it gives series in h... the hope is that it is convergent in ceratain sense and we can put $h=1$ and get the honest C^*-algebra. $\endgroup$ – Alexander Chervov Feb 13 '16 at 19:40

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