Given a locally compact quantum group \(\mathbb {G}\) and a dual unitary 2-cocycle \(\hat{\Omega }\) , any \(\hbox {W}^*\) -algebra A with a \(\mathbb {G}\) -action can be twisted into a new \(\hbox {W}^*\) -algebra \(A_{\hat{\Omega }}\) with an action by the cocycle twist \(\mathbb {G}_{\hat{\Omega }}\) of \(\mathbb {G}\) . We show how the standard space \(L^2(A_{\hat{\Omega }})\) , with its standard \(\mathbb {G}_{\hat{\Omega }}\) -representation, can be seen as a twist of \(L^2(A)\) with its standard \(\mathbb {G}\) -representation. Under an additional technical condition, we identify \(L^2(A_{\hat{\Omega }})\) with \(L^2(A)\) and describe the modular conjugation; we show that this special condition is satisfied when \(\hat{\Omega }\) arises from a skew bicharacter. We then apply these general results in the special case of generalized Drinfeld doubles and braided tensor products. As another application, we obtain a description of the universal \(\hbox {C}^*\) -algebra of functions on the generalized Drinfeld double.
