Spectral order of operators and range projections

We study the effect algebra (i.e. the positive part of the unit ball of an operator algebra) and its relation to the projection lattice from the perspective of the spectral order. A spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements. We show that if the spectral orthomorphism preserves the multiples of the unit, then it is a restriction of a Jordan homomorphism between the corresponding algebras. This is an optimal extension of the Dye's theorem on orthomorphisms of the projection lattices to larger structures containing the projections. Moreover, results on automatic countable additivity of spectral homomorphisms are proved. Further, we study the order properties of the range projection map, assigning to each positive contraction in a JBW algebra its range projection. It is proved that this map preserves infima of positive contractions in the spectral (respectively standard order) if, and only if, the projection lattice of the algebra in question is a modular lattice.