Deformation of operator algebras by Borel cocycles

Assume that we are given a coaction δ of a locally compact group G   on a C⁎C⁎-algebra A   and a TT-valued Borel 2-cocycle ω on G  . Motivated by the approach of Kasprzak to Rieffelʼs deformation we define a deformation AωAω of A  . Among other properties of AωAω we show that Aω⊗K(L2(G))Aω⊗K(L2(G)) is canonically isomorphic to A⋊δGˆ⋊δˆ,ωG. This, together with a slight extension of a result of Echterhoff et al., implies that for groups satisfying the Baum–Connes conjecture the K  -theory of AωAω remains invariant under homotopies of ω.