A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras

Let M be a von Neumann algebra with separating and cyclic vector ξ₀. The map $\text{xξ}{}_0\text{→ x}{}^1\text{ξ}{}_0$ with x?M has a least closed extension S. Tomita proved that the isometric involution J and the positive self-adjoint operator Δ obtained from the polar decomposition $\text{S = JΔ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of S satisfy JMJ = M′ and Δ^itMΔ^?it = M for any real t. More generally, he obtained similar results for the left von Neumann algebra of any generalized Hilbert algebra. In this paper a shorter proof of his results is given.