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Local Convergence in Measure on Semifinite von Neumann Algebras
A. M. Bikchentaev N. G. Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University
Abstract:
Suppose that
$\mathcal M$ is a von Neumann algebra of operators on a Hilbert space
$\mathcal H$ and
$\tau $ is a faithful normal semifinite trace on
$\mathcal M$. The set
$\widetilde {\mathcal M}$ of all
$\tau $-measurable operators with the topology
$t_{\tau }$ of convergence in measure is a topological
$*$-algebra. The topologies of
$\tau $-
local and
weakly $\tau $-local convergence in measure are obtained by localizing
$t_{\tau }$ and are denoted by
$t_{\tau \mathrm l}$ and
$t_{\mathrm w\tau \mathrm l}$, respectively. The set
$\widetilde {\mathcal M}$ with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in
$\widetilde {\mathcal M}$ with respect to the topologies
$t_{\tau \mathrm l}$ and
$t_{\mathrm w\tau \mathrm l}$ are proved. S.M. Nikol'skii's theorem (1943) is extended from the algebra
$\mathcal B(\mathcal H)$ to semifinite von Neumann algebras. The following theorem is proved: {\itshape For a von Neumann algebra
$\mathcal M$ with a faithful normal semifinite trace
$\tau $, the following conditions are equivalent\textup : \textup {(i)} the algebra
$\mathcal M$ is finite\textup ; \textup {(ii)}
$t_{\mathrm w\tau \mathrm l}= t_{\tau \mathrm l}$\textup ; \textup {(iii)} the multiplication is jointly
$t_{\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to
$\widetilde {\mathcal M}$\textup ; \textup {(iv)} the multiplication is jointly
$t_{\mathrm w\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to
$\widetilde {\mathcal M}$\textup ; \textup {(v)} the involution is
$t_{\tau \mathrm l}$-continuous from
$\widetilde {\mathcal M}$ to
$\widetilde {\mathcal M}$.}
UDC:
517.986+
517.987 Received in November 2005
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