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

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