SYMMETRIC BANACH ALGEBRAS OF OPERATORS IN A SPACE OF TYPE Π₁

Abstract

Some consequences of the usual neglect of transverse flow in this experiment have been examined. Expressions have been found for the ratio of the transmitted current to that which would flow across the apparatus with plane parallel geometry for particular models, numerical results given, and an estimate made for one of these of the corresponding error in the number of elastic collisions.

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

“…In particular, it shows that J = J if J is commutative. In order to do this we have to describe the canonical models of generic, uniformly closed algebras on Π 1 -spaces obtained in [10,11].…”

“…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.…”

“…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 such invariant subspaces is quite easy -it was shown in [6], that all of them (if weakly closed) are similar to W * -algebras and, therefore Theorem 1.1 clearly holds for them.…”

“…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.…”

“…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].…”