We generalized the C. R. Putnam theorem (1951), see also Problem 188 in the book (P. R. Halmos, A Hilbert space problem book. D. van Nostrand company, inc., London, 1967):.

Given a von Neumann algebra $M$ of operators on a Hilbert space $H$ and a faithful normal semifinite trace $\tau$ on $M$, denote by $S(M, \tau)$ the *-algebra of $\tau$-measurable operators. We obtain a sufficient condition ...