On freely acting automorphisms of \(C^*\)-algebras (Q790391)

Let (A,G,\(\alpha)\) be a \(C^*\)-dynamical system in which G is a discrete group with identity e. Let A'' be the enveloping von Neumann algebra of A. Then the action \(\alpha:g\mapsto \alpha_ g\) from G into the automorphism group Aut(A) of A induces the bitransposed action \(\alpha '':g\mapsto \alpha ''_ g\) Aut(A'') on A''. G is said to act centrally freely if \(\alpha\) ''\({}_ g\) acts freely on the centre of A'' for all \(g\neq e\); and G is said to act strongly centrally freely if there is a family \(\{p_ j\}\) of mutually orthogonal central projections in A'' such that \(\sum_{j}p_ j=1\) and \(\alpha_ g\!''(p_ j)p_ j=0\) for each j and \(g\neq e.\) The following results are announced. If G acts centrally freely and if A is a scattered \(C^*\)-algebra, then the \(C^*\)-crossed product \(A\times_{\alpha}G\) is scattered. If G acts strongly centrally freely and if A is a type I \(C^*\)-algebra, then \(A\times_{\alpha}G\) is also type I.