Weak expectations in \(C^*\)-dynamical systems (Q1120108)

\((A,G,\alpha)\) being a \(C^*\)-dynamical system and B an \(\alpha\)- invariant \(C^*\)-subalgebra of A, let \(({\mathcal H},\pi,u)\) be a covariant representation of \((B,G,\alpha)\) and \(M=(\pi(B)\cup u(G))''\). A covariant weak expectation is a completely positive linear contraction \(Q: A\to M\) such that \(Q| B=\pi\) and \(Q(\alpha_ t(a))=u_ tQ(a)u^*_ t\) where \(u\in A\) and \(t\in G\). Its existence is shown to be equivalent to the existence of a weak expectation from the \(C^*\)-crossed product \(A_{\alpha}\times G\) into M, i.e. a linear contraction \(\hat Q\) such that \(\hat Q(y)=(\pi \times u)\), \(y\in L^ 1(G,B)\). Analogous results are obtained for covariant representations associated to an \(\alpha\)- invariant state \(\phi\) of B. Moreover the existence of Q or \(\hat Q\) is shown to be asserted if \(\phi\) is G-central.