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.
