On the relative Dixmier property for inclusions of \(C^*\)-algebras
The relative Dixmier property for inclusions of \(C^\ast\)-algebras is investigated. An inclusion \(B\subset A\) of \(C^\ast\)-algebras with a conditional expectation \(E:A\to B\) is said to have the finite index if there exists \(c>0\) with \(E(x)>cx\) for all \(x\in A\). The relative Dixmier property of the inclusion \(B\subset A\) means that, for any \(x\in A\), the norm closed convex hull of the orbit of \(x\) with respect to the unitary group of \(B\), contains elements from the relative commutant of \(B\) in \(A\). In the \(C^\ast\)-context the Dixmier property for a single unital \(C^\ast\)-algebra \(B\) is defined as follows: For each \(x\in B\) the closed convex hull of the set \(\{uxu^\ast\mid u\in B \) unitary \(\}\) contains a multiple of the unit. It is shown that the Dixmier property of a unital simple algebra \(B\) admitting at least one tracial state implies already the relative Dixmier property of the inclusion \(B\subset A\). The proof is based on results on the weak relative Dixmier property for inclusions of von Neumann algebras and on relating the Dixmier property for \(C^\ast\)-algebras to the weak Dixmier property for von Neumann algebras arising naturally by the GNS representation. Also, the relative Dixmier property for finite von Neumann algebras is studied. As an application of the results obtained it is proved that if a simple \(C^\ast\)-algebra \(B\) has the Dixmier property then the crossed product of \(B\) by a discrete group \(G\) acting properly outer on it has the Dixmier property relative to \(B\). This gives a stronger version of Connes' local Rohlin's theorem for outer automorphisms on type \(\text{II}_1\) factors. Besides, the interplay between simplicity and the Dixmier property of \(C^\ast\)-algebras is analysed. The paper extends and simplify many results along this line.
outer automorphisms on type \(\text{II}_1\) factors
scientific article; zbMATH DE number 1423456
scientific article