A Radon-Nikodým theorem for natural cones associated with von Neumann algebras. II

Kosaki, Hideki

doi:10.1090/S0002-9939-1983-0681835-7

A Radon-Nikodým theorem for natural cones associated with von Neumann algebras. II
HTML articles powered by AMS MathViewer

by Hideki Kosaki PDF

Proc. Amer. Math. Soc. 87 (1983), 283-288 Request permission

Abstract:

A natural cone associated with a ($\sigma$-finite) von Neumann algebra is considered. Let ${\xi _0}$ be a cyclic and separating vector in the cone. For each vector $\xi$ in the cone, there always exists a positive selfadjoint operator $t$ affiliated with the algebra satisfying $\xi = tJtJ{\xi _0}$. Certain uniqueness results on $t$ for a given $\xi$ are also obtained.

References

Alberto Alonso and Barry Simon, The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory 4 (1980), no. 2, 251–270. MR 595414
Huzihiro Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309–354. MR 350437, DOI 10.2140/pjm.1974.50.309
Alain Connes, Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 4, x, 121–155 (1975) (French, with English summary). MR 377533
Uffe Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), no. 2, 271–283. MR 407615, DOI 10.7146/math.scand.a-11606
Hideki Kosaki, A Radon-Nikodým theorem for natural cones associated with von Neumann algebras, Proc. Amer. Math. Soc. 84 (1982), no. 2, 207–211. MR 637170, DOI 10.1090/S0002-9939-1982-0637170-5
F. Perdrizet, Éléments positifs relatifs à une algèbre hilbertienne à gauche, Compositio Math. 23 (1971), 25–47 (French). MR 285919
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Christian F. Skau, Positive selfadjoint extensions of operators affiliated with a von Neumann algebra, Math. Scand. 44 (1979), no. 1, 171–195. MR 544585, DOI 10.7146/math.scand.a-11801
M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168, DOI 10.1007/BFb0065832