  A Banach involutive algebra is called a Krein C*-algebra if there is a
fundamental symmetry (an involutive automorphism of period 2) such that the
C*-property is satisfied when the original involution is replaced with the new
one obtained by composing the automorphism with the old involution. For a given
fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even
part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part).
Our goal here is to develop a spectral theory for commutative unital Krein
C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the
even part and there exists an additional suitable "exchange symmetry" between
the odd and even parts.
