We introduce magnetic twisted actions of X = R n on general abelian C * -algebras and study the associated twisted crossed product and pseudodifferential algebras in the framework of strict deformation quantization.

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.

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Abstract. We show that Rieffel's deformation sends covariant C(T )-algebras into C(T )-algebras. We also treat the lower semi-continuity issue, proving that Rieffel's deformation transforms covariant continuous fields of C * -algebras into continuous fields of C * -algebras. Some examples are indicated, including certain quantum groups.

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