We define the decomposition property for partial actions of discrete groups on C * -algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems.Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups.

Abstract. We prove that, as in the case of global actions, any partial action gives rise to a groupoid provided with a Haar system, whose C * -algebra agrees with the crossed product by the partial action.

We introduce notions of weak and strong equivalence for nonsaturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional C * -algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

Abstract. With each Fell bundle over a discrete group G we associate a partial action of G on the spectrum of the unit fiber. We discuss the ideal structure of the corresponding full and reduced cross-sectional C * -algebras in terms of the dynamics of this partial action.

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