2018
Том 70
№ 8

All Issues

Muratov M. A.

Articles: 2
Article (Russian)

$(o)$-Topology in *-algebras of locally measurable operators

Chilin V. I., Muratov M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1531-1540

We consider the topology \( t\left( \mathcal{M} \right) \) of convergence locally in measure in the *-algebra \( LS\left( \mathcal{M} \right) \) of all locally measurable operators affiliated to the von Neumann algebra \( \mathcal{M} \). We prove that \( t\left( \mathcal{M} \right) \) coincides with the (o)-topology in \( L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} \) if and only if the algebra \( \mathcal{M} \) is σ-finite and is of finite type. We also establish relations between \( t\left( \mathcal{M} \right) \) and various topologies generated by a faithful normal semifinite trace on \( \mathcal{M} \).

Article (Russian)

Almost-Everywhere Convergence and (o)-Convergence in Rings of Measurable Operators Associated with a Finite von Neumann Algebra

Chilin V. I., Muratov M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1196-1205

We study the relationship between (o)-convergence and almost-everywhere convergence in the Hermite part of the ring of unbounded measurable operators associated with a finite von Neumann algebra. In particular, we prove a theorem according to which (o)-convergence and almost-everywhere convergence are equivalent if and only if the von Neumann algebra is of the type I.