Symmetry reduction of states I

In this article, 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, for which positive elements are sums of Hermitian squares a*a with a 2 A, but it 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 mu-red should be such that states on A mu-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, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a U(1)-action.