  We refer to the real Jordan Banach algebra of bounded Hermitian operators on
a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra
(AH-algebra) to be the directed group of an e-ring that contains a
semitransparent element, has the quadratic annihilation property, and satisfies
a Vigier condition on pairwise commuting ascending sequences. All of this
terminology is explicated in this article, where we launch a study of
AH-algebras. Here we establish the fundamental properties of AH-algebras,
including the existence of polar decompositions and spectral resolutions, and
we show that two elements of an AH-algebra commute if and only if their
spectral projections commute. We employ spectral resolutions to assess the
structure of maximal pairwise commuting subsets of an AH-algebra.
