A weak approximate innerness for abelian actions on \(C^*\)-algebras (Q1180532)

A strongly continuous one-parameter automorphism group \(\alpha\) of a \(C^*\)-algebra \(A\) is called approximately inner if there is a sequence \(\{h_ n\}\) of self-adjoint elements of \(A\) such that \(\alpha_ t(x)=\lim_{n\to\infty}\exp(ith_ n)x exp(-ith_ n)\), \(x\in A\), uniformly on every compact subset of \(t\in\mathbb{R}\). In the present paper the author introduced a concept of weak approximate innerness: For each \(t\in\mathbb{R}\) there is a sequence \(\{b_ n\}\subseteq A\) such that \(\| b_ n x-\alpha_ t(x)b_ n\|\to 0\), \(\| b_ n\|\to 1\), \(\| b_ n x\|\to\| x\|\) and \(Sp_ \alpha(b_ n)\to\{0\}\) for any \(x\) in \(A\), where \(Sp_ \alpha(b_ n)\) denotes the \(\alpha\)- spectrum of \(b_ n\). It is shown that under some conditions the former innerness does imply the latter, and there exists an example satisfying the latter but not the former. The main result states that if \((A,G,\alpha)\) is a \(C^*\)-dynamical system where \(A\) is separable, \(G\) is locally compact and Abelian with countable basis, and \(\alpha\) is an action of \(G\) on \(A\), then \(\alpha\) is weak approximately inner if and only if there exists a faithful family of coherent irreducible representations of \((A,G,\alpha)\).