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Induced Mappings On Boolean Algebras Of Clopen Sets And On Projections Of The C^*-Algebra C(X), Ahmed Al-Rawashdeh, Wasfi Shatanawi

Turkish Journal of Mathematics

For a compact space X, any group automorphism \varphi of C(X,S^1) induces a mapping \Theta on the Boolean algebra of the clopen subsets of X. We prove that the disjointness of \Theta equivalent to \theta_{\varphi} is an orthoisomorphism on the sets of projections of the C^*-algebra C(X), when \varphi(-1)=-1. Indeed, \Theta is a Boolean isomorphism iff \theta_{\varphi} preserves the product of projections. If X is equipped with a probability measure \mu, on a certain \sigma-algebra of X, we show (under some condition) that \Theta preserves the disjoint of clopen subsets, up to sets of measure zero, or equivalently, the mapping …