, Volume 3, Issue 4, pp 718-765

Covariance algebra of a partial dynamical system

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Abstract

A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C *-dynamical systems.

In this paper we construct and investigate a universal C *-algebra C *(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto.

The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system, \((\tilde X,\tilde \alpha )\) where \(\tilde \alpha \) is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C *(X, α).