Symmetry Reduction of States I

We develop a general theory of symmetry reduction of states on (possibly non-commutative) \*-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra $g$. The key idea advocated for in this article is that the "correct" notion of positivity on a \*-algebra $A$ is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares $a^\* a$ with $a \in A$, but can be a more general one that depends on the example at hand, like pointwise positivity on \*-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on $A$ thus depends on this choice of positivity on $A$, and the notion of positivity on the reduced algebra $A\_{red}$ should be such that states on $A\_{red}$ are obtained as reductions of certain states on $A$. We discuss three examples in detail: Reduction of the \*-algebra of smooth functions on a Poisson manifold $M$, which reproduces the coisotropic reduction of $M$; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.