Equivalence of order and algebraic properties in ordered $^*$-algebras

Abstract

The aim of this article is to describe a class of $^*$-algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered $^*$-algebras ($^*$-algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a $C^*$-norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered $^*$-algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su $^*$-algebras as uniformly complete Archimedean ordered $^*$-algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su $^*$-algebras generalize some important properties of $C^*$-algebras to algebras of unbounded operators. Similarly, they generalize some properties of $\varPhi$-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered $^*$-algebras. As an example, Su $^*$-algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as $^*$-algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system.