Reduction of Lie-Jordan Banach algebras and quantum states

TL;DR

This paper develops a reduction theory for Lie-Jordan Banach algebras, linking it to quantum constraints and C*-algebra reductions, and characterizes the state space and GNS representations of the reduced algebras.

Contribution

It introduces a new reduction framework for Lie-Jordan Banach algebras and relates it to standard quantum observable reductions, providing a unified approach.

Findings

Reduction of Lie-Jordan Banach algebras with respect to ideals or subalgebras.

Connection established between this reduction and standard C*-algebra quantum constraints.

Characterization of state spaces and GNS representations of reduced algebras.

Abstract

A theory of reduction of Lie-Jordan Banach algebras with respect to either a Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared with the standard reduction of C*-algebras of observables of a quantum system in the presence of quantum constraints. It is shown that the later corresponds to the particular instance of the reduction of Lie-Jordan Banach algebras with respect to a Lie-Jordan subalgebra as described in this paper. The space of states of the reduced Lie-Jordan Banach algebras is described in terms of equivalence classes of extensions to the full algebra and their GNS representations are characterized in the same way. A few simple examples are discussed that illustrates some of the main results.