Infinitesimal generators of \(C^*\)-algebras (Q679207)

Let \(G\) be a locally compact group, and let \(C^*_r(G)\) be its reduced \(C^*\)-algebra acting on \(L^2(G)\) by left translations. Extending a method introduced for discrete groups by \textit{E. C. Lance} and himself in [Proc. Am. Math. Soc. 61, 310-314 (1976; Zbl 0345.46046)], the author associates to every continuous homomorphism from \(G\) to \(\mathbb{R}\) a derivation which is an infinitesimal generator of a one-parameter group of automorphisms of \(C^*_r(G)\). In case \(G\) is discrete and \(\Hom (G,\mathbb{R})\neq 0\), he shows that, for any \(C^*\)-algebra \(A\), there exists a one-parameter group of automorphisms of the spatial tensor product \(C^*_r(G) \otimes A\) that is not approximately inner.