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

Abstract

Mathematics Subject Classification: 46K15, 47B50, 46L10We 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 J with identity on a Π 1 -space H which has a cyclic and separating vector, there is an antilinear J-involution j : H → H such that jJ j = J .The paper also contains a full proof of the Double Commutant Theorem for J-symmetric operator algebras on Π 1 -spaces.Keywords and phrases :J-symmetric operator algebras on Pontryagin Π 1 -spaces, quasivectors on operator * -algebras, Tomita-Takesaki theory for quasivectors, Double Commutant Theorem. * The author is grateful to the University of North London for inviting him to Britain for a threemonth research visit. . Downloaded from www.worldscientific.com by UNIVERSITY OF TOKYO on 06/07/15. For personal use only.V. S. SHULMAN Theorem 1.1. For any weakly closed J-symmetric operator algebra J with identity on a Π 1 space H which has a cyclic and separating vector, there is an antilinear J-involution j : H → H such that jJ j = J .An indefinite scalar product with a finite rank K of indefiniteness turns a linear space into a Pontrjagin's space Π K . In other words, Π K -spaces are indefinite metric spaces which decompose into the direct sum of a Hilbert space and a K-dimensional "negative" space. For operator algebraists, Π K -spaces are attractive primarily because of the fact that the theory of J-symmetric operator algebras on these spaces combines, in a natural way, two well-developed theories: C * -algebras and operator algebras on finite-dimensional spaces. From the first works of Naimark (see, for example, [8]) and Ismagilov [4] it became clear that quite apart from deep intrinsic problems this theory sheds new light on and raises interesting new questions about self-adjoint operator algebras on Hilbert spaces. To see the importance of Π K -spaces for physical applications it suffices to take into account, that all irreducible representations of Lorentz group can be realised as J-unitary (preserving indefinite metrics) representations on these spaces.Our approach to modular theory is based on the classification (division into several types and construction of canonical models for every type) of J-symmetric operator algebras on Π 1 -spaces, obtained in [11]. For this purpose, essential development and refinement were done of the main technical tool in [11] -the theory of quasivectors of operator * -algebras on Hilbert spaces. In this way we obtain the precise description of commutants of canonical models and give the full proof of the Double Commutant Theorem for J-symmetric operator agebras on Π 1 -spaces, outlined in [11].In what follows only generic algebras are considered, that is, the algebras which have at least one neutral invariant subspace. The case of algebras without ...