Crossed products for actions of crossed modules on [C.sup.]-algebras

We decompose the crossed product functor for actions of crossed modules of locally compact groups on [C.sup.*]-algebras into more elementary constructions: taking crossed products by group actions and fibres in [C.sup.*]-algebras over topological spaces. For this, we extend Takesaki-Takai duality to Abelian crossed modules; describe the crossed product for an extension of crossed modules; show that equivalent crossed modules have equivalent categories of actions on C*-algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules. Mathematics Subject Classification (2010). 46L55, 18D05. Keywords. [C.sup.*]-algebra, crossed module, Fell bundle, Takesaki-Takai duality.