Partially inner \(C^*\)-dynamical systems
Let \((A,G,\alpha)\) be a \(C^*\)-dynamical system with G a locally compact abelian group. The system is called H-inner for some closed subgroup H of G, if there is a strictly continuous unitary representation v of H into the algebra \(M(A)^{\alpha}\) of \(\alpha\)-invariant multipliers of A, such that \(\alpha_ t=Ad v_ t\) for every t in H. In this case one can form the restricted (or twisted) crossed product \(G\times_{\alpha | H}A\) in the sense of Dang Ngoc and P. Green, and find a surjective morphism \(\Phi\) from the ordinary crossed product \(G\times_{\alpha}A\) onto \(G\times_{\alpha | H}A\). The restriction to the subgroup \(H^{\perp}\) of \(\hat G\) of the dual action \({\hat \alpha}\) on \(G\times_{\alpha}A\) is invariant under \(\Phi\), so that we obtain a \(C^*\)-dynamical system \((G\times_{\alpha | H}A,H^{\perp},{\hat \alpha})\). We induce this system to obtain the \(C^*\)-algebra \[ Ind^{\hat G}_{H^{\perp}}(G\times_{\alpha | H}A). \] The main result (Theorem 2.4) is that the map \[ \Psi:G\times_{\alpha}A\to Ind^{\hat G}_{H^{\perp}}(G\times_{\alpha | H}A) \] defined by \(\Psi (x)(\sigma)=\Phi ({\hat \alpha}_{-\sigma}(x))\) is an isomorphism. If the \(C^*\)-algebra \(A\subset B({\mathcal H})\), the algebra of all bounded operators on a Hilbert space \({\mathcal H}\), then there is a faithful ''regular'' representation \(\rho\) of \(G\times_{\alpha | H}A\) on the induced Hilbert space \(Ind^ G_ H({\mathcal H})\). This in turn leads to a representation \({\tilde \rho}\) of the induced \(C^*\)-algebra above on the Hilbert space \(Ind^{\hat G}_{H^{\perp}}(Ind^ G_ H({\mathcal H})),\) which is the image under a Plancherel-type isometry w of \(L^ 2(G,{\mathcal H})\). Now \({\tilde \rho}\circ \Psi =Ad w\); and having the isomorphism \(\Psi\) in spatial form like this, allows an extension of the \(C^*\)-results to H-inner \(W^*\)-dynamical systems.
strictly continuous unitary representation
scientific article; zbMATH DE number 3955752
scientific article