Several variants of the
classical Gelfand-Neumark characterization of complex C∗-algebras are here extended
to characterize real C∗-algebras up to isometric*-isomorphism and also up to
homeomorphic ∗-isomorphism. The proofs depend on norming the complexification of
the real algebra and applying the author’s characterization of complex C∗-algebras to
the result. L. Ingelstam has obtained similar but weaker results by an entirely
different method.