Quasivectors and Tomita–Takesaki Theory for Operator Algebras on Π1-Spaces

Abstract

We consider operator algebras, which are symmetric with respect to an indefinite scalar product. It is shown, that in the case when the rank of indefiniteness is equal to 1 there exists a working modular theory, and in particular a precise analogue of the Fundamental Tomita's Theorem holds: For any weakly closed J-symmetric operator algebra [Formula: see text] with identity on a Π1-space H which has a cyclic and separating vector, there is an antilinear J-involution j : H→H such that [Formula: see text]. The paper also contains a full proof of the Double Commutant Theorem for J-symmetric operator algebras on Π1-spaces.