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We introduce an axiomatization of the notion of a semidirect product oflocally compact quantum groups and study properties. Our approach is slightlydifferent from the one introduced in the thesis of S.~Roy and, unlike theinvestigations of Roy, we work within the von Neumann algebraic picture. Thisallows the use of powerful techniques related to crossed products by actions oflocally compact quantum groups. In particular we show existence of a "braidedcomultiplication" on the algebra spanned by slices of "braided multiplicativeunitary".

Quantum groups with projection on von Neumann algebra level