Twisted dual-group algebras: Equivariant deformations of \(C_ 0(G)\)
``We consider a family of twisted Fourier algebras \(A(G, \omega)\) of a locally compact group \(G\), which in the case of an abelian group \(G\) are the Fourier transforms of the usual twisted group algebras of \(\widehat G\). The corresponding \(C^*\)-algebras \(C^* (\widehat G, \omega)\) are deformations of \(C_0 (G)\), which are equivariant in the sense that \(G\) still acts by left translation. The main examples come from cocycles \(\sigma\) on the dual of an abelian subgroup \(H\) of \(G\); we prove that for such cocycles the twisted dual-group algebras \(C^* (\widehat G, \omega)\) are induced from the twisted group algebras \(C^* (\widehat H, \sigma)\), and we give detailed formulas for the multiplication on \(A(G, \omega)\) which extend to larger dense subalgebras of \(C_0 (G)\) and \(C_b (G)\). We anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces \(C_0 (G/ \Gamma)\).'' The formulation goes like this. Let \({\mathcal L} (G)\) be the group von Neumann algebra (generated by the left regular representation), and let \(\delta_G : {\mathcal L} (G) \to {\mathcal L} (G) \otimes {\mathcal L} (G)\) be the unique normal homomorphism such that \(\delta_G (L_x) = L_x \otimes L_x\) on the left translation operators. Then a dual cocycle is a unitary \(\omega \in {\mathcal L} (G) \otimes {\mathcal L} (G)\) satisfying \[ \omega_{12} \bigl( \delta_G \otimes i (\omega) \bigr) = \omega_{23} \bigl( i \otimes \delta_G (\omega) \bigr) \quad \text{in } {\mathcal L} (G) \otimes {\mathcal L} (G) \otimes {\mathcal L} (G) \] (which, for abelian \(G\), is the image of the cocycle identity under the Fourier transform of \({\mathcal L} (G)\) onto \(L^\infty (\widehat G))\). For \(f,g \in A (G)\), the twisted multiplication \(f*_\omega g\) in the Fourier algebra \(A(G)\) is defined by \[ \langle T, f*_\omega g \rangle = \bigl \langle \delta_G (T) \omega^*, f \otimes g \bigr \rangle \quad \text{for } T \in {\mathcal L} (G). \]
scientific article; zbMATH DE number 790374