Martingale-type convergence of modular automorphism groups on von Neumann algebras (Q796795)

Given a faithful semifinite normal weight \(\phi\) on a von Neumann algebra M, let \(\Delta_{\phi}\) and \(\sigma_ t^{\phi}\) be the associated modular operator and modular automorphism group. In this paper, the strong convergence of modular automorphism groups \(\sigma_ t^{\phi_{\alpha}}\) associated with \(\phi_{\alpha}=\phi \upharpoonright M_{\alpha}\) is considered for increasing (resp. decreasing) von Neumann subalgebras \(M_{\alpha}\nearrow M\) (resp. \(M_{\alpha}\searrow M_{\infty}).\) For the increasing case, it is proved that \(\sigma_ t^{\phi_{\alpha}}(x)\to \sigma_ t^{\phi}(x)\) strongly for every \(x\in \cup_{\alpha}M_{\alpha}\) if and only if the union of the left Hilbert algebras associated with \(\phi_{\alpha}\) is a core of \(\Delta_{\phi}^{1/2}\). For the decreasing case, it is proved under suitable assumptions that \(\sigma_ t^{\phi_{\alpha}}(x)\to \sigma_ t^{\phi_{\infty}}(x)\) strongly for every \(x\in M_{\infty}\) where \(\phi_{\infty}=\phi \upharpoonright M_{\infty}.\) The strong convergence of Connes cocycle derivatives is obtained in the similar situation of increasing or decreasing von Neumann subalgebras. Finally some discussions are given in connection with the martingale convergence in von Neumann algebras. In this direction, further discussions have been given in \textit{F. Hiai} and \textit{M. Tsukada} [Trans. Am. Math. Soc. 282, 791-798 (1984)].