Convergences in \(W^ *\)-algebras (Q1088130)

Convergences closely on large sets (c.l.s.), nearly everywhere (n.e.), almost uniformly (a.u.) and quasi-uniformly (q.u.) for a sequence of observables from a \(W^*\)-algebra \({\mathfrak A}\) with a faithful normal state \(\rho\) are all of ''a.e.-type''. Namely, in the commutative case, where \({\mathfrak A}=L^{\infty}(\Omega,\mu)\) and \(\rho (x)=\int xd\mu\), each of these types of convergence is equivalent to the \(\mu\)-a.e. one in \(\Omega\). The main theorem of the paper states, in particular, that the q.u. convergence implies a.u. one, a.u. implies c.l.s., c.l.s. is equivalent to n.e. Moreover, they are all equivalent for a bounded sequence of operators and, in general, they are not equivalent for unbounded sequences. For convergence of subsequences, see the paper ''Convergences almost everywhere in \(W^*\)-algebras'' by the author in Lect. Notes Math. 1136, 420-427 (1985; Zbl 0576.46047).