Статья
2013
The Haagerup problem on subadditive weights on W*-algebras. II
A. M. Bikchentaev
Русская математика
https://doi.org/10.3103/S1066369X13120074
Abstract / Full Text
In 1975 U. Haagerup has posed the following question: Whether every normal subadditive weight on a W*-algebra is σ-weakly lower semicontinuous? In 2011 the author has positively answered this question in the particular case of abelian W*-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional W*-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic W*-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic W*-algebras.
Author information
- Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008, RussiaA. M. Bikchentaev
References
- U. Haagerup, “Normal Weights on W*-Algebras,” J. Funct. Anal. 19(3), 302–317 (1975).
- A.M. Bikchentaev, “The Haagerup Problemon Subadditive Weights on W*-Algebras,” Izv. Vyssh. Uchebn. Zaved. Mat., 2011, No. 10, 94–98 (2011) [Russian Mathematics (Iz. VUZ) 55 (10), 82–85 (2011)].
- M. Takesaki, Theory of Operator Algebras (Springer-Verlag, New York-Heidelberg-Berlin, 1979).
- O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1. C*- and W*-Algebras, Symmetry Groups, Decomposition of States (Texts and Monographs in Physics, Springer-Verlag, New York-Heidelberg-Berlin, 1979).
- J. Dixmier, “Existence de Traces non Normales,” C. R. Acad. Sci., Paris, Sér. A 262, 1107–1108 (1966).
- A. L. Carey and F. A. Sukochev, “Dixmier Traces and Some Applications in Non-Commutative Geometry,” Usp. Mat. Nauk 61(6), 45–110 (2006).
- I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators (Nauka, Moscow, 1965; American Mathematical Society, Providence, RI, 1969).
- A. M. Bikchentaev, “Block Projection Operators in Normed Solid Spaces of Measurable Operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 86–91 (2012) [Russian Mathematics (Iz. VUZ) 56 (2), 75–79 (2012)]
- A. M. Bikchentaev, “Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras,” Mat. Zametki 75(3), 342–349 (2004).
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, 4th Ed. (Nevskii Dialekt, St.-Petersburg, 2004).
- A. M. Bikchentaev, “On a Property of L p Spaces on Semifinite von Neumann Algebras,” Mat. Zametki 64(2), 2, 185–190 (1998).