Induced mappings on Boolean algebras of clopen sets and on projections of the $C^{ast}$ - Algebra C (X)

Abstract

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 $C^{ast}$−algebra C(X), when $varphi$(−1) = −1. Indeed,$Theta$ is a Boolean isomorphism if $theta_{varphi}$ 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 $theta_{varphi}$ is $mu$−orthoisomorphism on the projections of the $C^{ast}$−algebra C(X).