Induced mappings on boolean algebras of clopen sets and on projections of the C*-algebra C(X)

For a compact space X, any group automorphism ψ of C(X,double struk D sign¹) induces a mapping Θ on the Boolean algebra of the clopen subsets of X. We prove that the disjointness of Θ equivalent to θ_ψ, is an orthoisomorphism on the sets of projections of the C*-algebra C(X), when ψ(-1) = -1. Indeed, Θ is a Boolean isomorphism iff θ_ψ preserves the product of projections. If X is equipped with a probability measure μ, on a certain σ-algebra of X, we show (under some condition) that Θ preserves the disjoint of clopen subsets, up to sets of measure zero, or equivalently, the mapping θ_ψ is μ-orthoisomorphism on the projections of the C*-algebra C(X).