{1m(A) - i} -1 E (SB(H), SOT) are continuous. (H) with respect to the strong resolvent topology if and only if {Re(A",)-i}-l~ ----; {Re(A)-i}-l~, {1m(A",)-i}-l~ ----; {1m(A)-i}-l~, for each ~ E H. This topology is well-studied in the field of unbounded operator theory and suitable for the operator theoretical study. We denote the system of open sets of the strong resolvent topology by OSRT. Let 9J1 be a finite von Neumann algebra on a Hilbert space H.

5. 6. 7. 8. 9. 10. Lie Group-Lie Algebra Correspondences of Unitary Groups in Finite von Neumann Algebras, submitted. 4850 H. Ando, Y. Matsuzawa and I. Ojima, in preparation D. Beltita, Lie theoretic significance of the measure topologies associated with a finite trace, Forum Math. 22 (2010), 241-253. T. Fack and H. Kosaki, Generalized s-numbers of T-measurable operators, Pacific J. , 123 (1986), 260-300. K. Hofmann and S. Morris, The Lie Theory of Connected Pro-Lie Groups, Europ. Math. Soc. Publ.

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