# Symmetry Reduction of States I

### Philipp Schmitt

Leibniz Universität Hannover, Germany### Matthias Schötz

Instytut Matematyczny PAN, Warsaw, Poland

## Abstract

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 $a∈A$, 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.

## Cite this article

Philipp Schmitt, Matthias Schötz, Symmetry Reduction of States I. J. Noncommut. Geom. 18 (2024), no. 2, pp. 501–545

DOI 10.4171/JNCG/534