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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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