Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action \alpha of a group G on an associative algebra A we consider the crossed product A x_\alpha G. Using the algebras of multipliers of ideals of A we prove that A x_\alpha G is associative, provided that all ideals of A are idempotent. This generalizes a previous result on the associativity of A x_\alpha G in the context of C*-algebras. We also give a criteria for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with crossed product structure.