  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 Phi-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.
