The Crossed Product by a Partial Endomorphism

Abstract

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I , a *-homomorphism α : A → M(I ) and a map L : J → A with some properties, based on earlier works of Pimsner and Katsura, we define a C*-algebra $$ \mathcal{O}{\left( {A,\alpha ,L} \right)} $$ which we call the Crossed Product by a Partial Endomorphism. We introduce the Crossed Product by a Partial Endomorphism $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ induced by a local homeomorphism σ : U → X where X is a compact Hausdorff space and U is an open subset of X. A bijection between the gauge invariant ideals of $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ and the σ, σ-1- invariant open subsets of X is showed. If (X, σ) has the property that $$ {\left( {{X}\ifmmode{'}\else$'$\fi,\sigma _{{\left| {{X}\ifmmode{'}\else$'$\fi} \right.}} } \right)} $$ is topologically free for each closed σ, σ-1-invariant subset X′ of X then we obtain a bijection between the ideals of $$ \mathcal{O}{\left( {X,\alpha ,L} \right)} $$ and the open σ, σ-1-invariant subsets of X.