Journals / Turkish Journal of Mathematics / 2007 / Cilt: 31 - Sayı: 4
Induced Mappings on Boolean Algebras of Clopen Sets and on Projections of the C*-Algebra C(X)
- Pages
- 439–451
- DOI
- —
Abstract
For a compact space X, any group automorphism j of C(X,S1) induces a mapping Q on the Boolean algebra of the clopen subsets of X. We prove that the disjointness of Q equivalent to qj is an orthoisomorphism on the sets of projections of the C*-algebra C(X), when j(-1)=-1. Indeed, Q is a Boolean isomorphism iff qj preserves the product of projections. If X is equipped with a probability measure m, on a certain s-algebra of X, we show (under some condition) that Q preserves the disjoint of clopen subsets, up to sets of measure zero, or equivalently, the mapping qj is m-orthoisomorphism on the projections of the C*-algebra C(X).
Özet
For a compact space X, any group automorphism j of C(X,S1) induces a mapping Q on the Boolean algebra of the clopen subsets of X. We prove that the disjointness of Q equivalent to qj is an orthoisomorphism on the sets of projections of the C*-algebra C(X), when j(-1)=-1. Indeed, Q is a Boolean isomorphism iff qj preserves the product of projections. If X is equipped with a probability measure m, on a certain s-algebra of X, we show (under some condition) that Q preserves the disjoint of clopen subsets, up to sets of measure zero, or equivalently, the mapping qj is m-orthoisomorphism on the projections of the C*-algebra C(X).