The singular value function, associated with a Maharamtrace
von Neumann algebra, algebra of measurable operators,vector-valued trace, Dedekind complete vector lattice, singular value function, Maharam trace.
DOI:
https://doi.org/10.56143/ujmcs.v1i1.6Ключевые слова:
von Neumann algebra, algebra of measurable operators, vector-valued trace, Dedekind complete vector lattice, singular value function, Maharam trace.Аннотация
Let $M$ be a finite von Neumann algebra, let $S(M)$ be the $*$-algebra of measurable operators affiliated with $M$. Maharam traces $\Phi$ on a von Neumann algebra $M$ with values in complex Dedekind complete vector lattices are considered. The singular value function of operators from $S(M)$, associated with such a trace $\Phi$ are determined. The main properties of these singular value functions, similar to classical singular value functions of measurable operators, are studied.
Библиографические ссылки
[1] Segal I.E. A non-commutative extension of abstract integration, Ann. Math. – 1953. – 57. – P. 401–457.
[2] Dixmier J. Formes lineares sur un anneau d’operators. Bull. de la Soc. Math. de France 1953, V. 81, 9–39.
[3] Nelson E. Notes on non commutative integration J. Funct. Anal. 1974. No 15. p. 103–116.
[4] Yeadon F.J. Convergence of measurable operators Proc. Camb. Phil. Soc. – 1973. – V. 74. – P. 257–268.
[5] Yeadon F.J. Non-commutative Lp-spaces Math. Proc. Camb. Phil. Soc. – 1975. – V. 77. – P. 91–102.
[6] Fack T., Kosaki H. Generalised s-numbers of t-measurable operators Pacif. J. Math. 123 (1986), 269–300.
[7] Dixmier J. Les algebras d’operateurs dans l’espace Hilbertien (Algebres de von Neunann). Paris: Gauthier-Villars, 1969. –367 p.
[8] Stratila S., Zsido L. Lectures on von Neumann algebras. – England Abacus Press, 1975. –477 p.
[9] Takesaki M. Theory of operator algebras I. – New York: Springer, 1979. –415 p.
[10] Muratov M.A., Chilin V.I. Algebras of measurable and locally measurable operators. Kyiv, Pratsi In-ty matematiki NAN
Ukraini. 2007. V. 69. 390 p. (Russian).
[11] Dodds P.G., Ben de Pagter, Sukochev F.A. Noncommutative integration and operator theory. Progress in Nathematics 349, Birkhauser, 2023, 577 p.
[12] Kusraev A.G., Dominanted Operators, Mathematics and its Applications, 519, Kluwer Academic Publishers, Dordrecht, 2000. 446 p.
[13] Chilin V., Zakirov B. Maharam traces on von Neumann algebras Methods of Functional Analysis and Topology Vol. 16 (2010), 2. pp. 101–111.
[14] Zakirov B.S., Chilin V.I. Noncommutative integration for traces with values in Kantorovich-Pinsker spaces Russian Math. (Iz. VUZ), 2010, 54(10), 15-26.
[15] Chilin V., Zakirov B. Non-commutative Lp -spaces associated with Maharam trace J. Operator theory, 2012, 68(1), 67–83.
[16] Muratov M.A., Chilin V.I. Central extensions of *-algebras of measurable operators Dokl. NAN Ukraini, Kyiv, 2009. 7. –P. 24–28. (Russian).
[17] Zakirov B.S., Chilin V.I. Duality for non-commutative L1-spaces, associated with Maharam trace Research in Mathematical Analysis. Series Mathematical Forum. V.3. VSC RAS, Vladikavkaz, 2009, 67–91.
