Type III representations and automorphisms of some separable nuclear \(C^*\)-algebras. (Q1869062)

Let \(A\) be a nuclear separable \(C^*\)-algebra and let \(\pi_1,\pi_2\) be its representations on a separable Hilbert space such that \(\pi_1(A)''=\pi_2(A)''={\mathcal M}\). The main result is the equivalence of the following properties: (i) There is a sequence \(\{U_n\}\) of unitaries in \({\mathcal M}\) such that \(\lim_n \text{Ad}_{U_n}\pi_1(x)-\pi_2(x)=0\) for any \(x\in A\); (ii) there is an asymptotically inner automorphism \(\alpha\) of \(A\) such that \(\pi_1\circ\alpha\) and \(\pi_2\) are equivalent.