Covariance algebra of a partial dynamical system

By a partial dynamical system we mean a pair (X,α) where X is compact and α : ∆ → X is a continuous, not necessarily injective, mapping such that ∆ is clopen, additionally we assume that α(∆) is open. Such systems arise naturally when dealing with commutative C∗-dynamical system (A, δ). We associate with every pair (X,α), or (A, δ), a covariance C∗-algebra C∗(X,α) = C∗(A, δ) which agrees with a partial crossed product [7] in case α is injective, and a crossed product by a monomorphism [17] in case α is onto. The main idea is to profit from a description of maximal ideal space of a coefficient algebra, cf. [12], [13], which enables us to construct the reversible extension of (X,α), i.e. a larger system (X̃, α̃) where α̃ is a partial homeomorphism, and hence apply partial crossed product theory [7], [10]. Moreover, we investigate the relevance between (X,α) and C∗(X,α). We generalize the notions of topological freedom and invariance of a set, which we use to obtain isomorphism theorem and description of ideals of C∗(X,α). AMS Subject Classification: 47L65, 46L45, 37B99