Reduction of Lie–Jordan Banach algebras and quantum states

Abstract

In this paper, it is shown that the concept of dynamical correspondence for Jordan Banach algebras is equivalent to a Lie structure compatible with the Jordan one. Then a theory of reduction of Lie–Jordan Banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of C*-algebras of observables of a quantum system. The space of states of the reduced Lie–Jordan Banach algebra is characterized in terms of Dirac states on the physical algebra of observables and its GNS representations described in terms of states on the unreduced algebra.