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Divisible operators in von Neumann algebras. (English) Zbl 1236.47033
Let \(M\) be a von Neumann algebra and \(x\in M\). Let \(n\) be a cardinal larger than one and \(M_n\) denote a type \(I_n\) factor. Then \(x\) is said to be \(n\)-divisible if the relative commutant \(W^*(x)'\cap M\) unitally contains the factor \(M_n\). A *-homomorphism \(\rho\) between two \(C^*\)-algebras is said to be \(n\)-divisible if its relative commutant unitally contains the factor \(M_n\). In this article, a basic question is: when is the set of \(n\)-divisible operators in \(M\) dense in \(M\) by given different topologies? By considering various operator topologies, such as the \({\sigma}\)-weak, \({\sigma}\)-strong, \({\sigma}\)-strong* topology, the author decides whether the sets of \(n\)-divisible operators are dense or not. It is worthy to mention that, in the case of the \({\sigma}\)-strong topology, the density can be determined by assuming that \(M\) is McDuff (i.e., \(M\cong M\overline{\otimes} R\)).
Let \(U(x)\) denote the unitary orbit of \(x\in M\) and \(\overline{U(x)}^{\sigma -w}\) denote the \(\sigma\)-weakly closure of \(U(x)\). The author also proves that, for any \(x\) belonging to the norm closure of the \(\aleph_0\)-divisible operators, \(\overline{U(x)}^{\sigma -w}\) is convex. Some techniques and facts used here are also related to some useful concepts in the \(K\)-theory of \(C^*\)-algebras.

MSC:
47C15 Linear operators in \(C^*\)- or von Neumann algebras
47A65 Structure theory of linear operators
46L10 General theory of von Neumann algebras
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