We consider a magnetic Schr\"odinger operator $H^h$, depending on the
semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We
assume that there is no electric field. We suppose that the minimal value $b_0$
of the magnetic field $b$ is strictly positive, and there exists a unique
minimum point of $b$, which is non-degenerate. The main result of the paper is
a complete asymptotic expansion for the low-lying eigenvalues of the operator
$H^h$ in the semiclassical limit.

First, we review the notion of a Poisson structure on a noncommutative
algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian
vector field on a noncommutative Poisson algebra. Then we describe a Poisson
structure on a noncommutative algebra associated with a transversely symplectic
foliation and construct a class of Hamiltonian vector fields associated with
this Poisson structure.