  The classical Gelfand--Naimark theorems provide important insight into the
structure of general and of commutative C*-algebras. It is shown that these can
be generalized to certain ordered *-algebras. More precisely, for
$\sigma$-bounded closed ordered *-algebras a faithful representation as
operators is constructed. Similarly, for commutative such algebras, a faithful
representation as complex-valued functions is constructed if an additional
necessary regularity condition is fulfilled. These results generalize the
Gelfand--Naimark representation theorems to classes of *-algebras larger than
C*-algebras, and which especially contain *-algebras of unbounded operators.
The key to these representation theorems is a new result for Archimedean
ordered vector spaces V: If V is $\sigma$-bounded, then the order of V is
induced by the extremal positive linear functionals on V.
