Abstract
In this paper we consider traces on a von Neumann algebra M with values in complex Kantorovich-Pinsker spaces. We establish the connection between the convergence with respect to the trace and the convergence locally in measure in the algebra S(M) of measurable operators affiliated with M. We define the (bo)-complete lattice-normed spaces of integrable operators in S(M) and prove that they are decomposable if the trace possesses the Maharam property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. E. Segal, “A Non-Commutative Extension of Abstract Integration,” Ann.Math. (2) 57, 401–457 (1953).
E. Nelson, “Notes on Non-Commutative Integration,” J. Funct. Anal. 15, 103–116 (1974).
F. J. Yeadon, “Non-Commutative L p-Spaces,” Math. Proc.Camb. Phil.Soc. 77, 91–102 (1975).
T. Fack and H. Kosak, “Generalized s-Numbers of τ-Measurable Operators,” Pacif. J. Math. 123, 269–300 (1986).
G. Pisier and Q. Xu, “Non-Commutative L p-Spaces,” in: Handbook of the Geometry of Banach Spaces (North-Holland, Amsterdam, 2003), Vol. 2, pp. 1459–1517.
A. G. Kusraev, Majorized Operators (Nauka, Moscow, 2003) [in Russian].
I. G. Ganiev and V. I. Chilin, “Measurable Bundles of Noncommutative L p-Spaces Associated With a Center-Valued Trace,” Matem. Trudy 4(2), 27–41 (2001).
V. I. Chilin and A. A. Katz, “On Abstract Characterization of Non-Commutative L p-Spaces Associated with a Center-Valued Trace,” Methods Funct. Anal. Topol. 11(4), 346–355 (2005).
S. Stratila and L. Zsido, Lectures on von Neumann Algebras (Editura Academici Republcii Socialiste Romania, Bucuresti, 1975).
M. Takesaki, Theory of Operator Algebras, Vol. I (Springer, New York, Heidelberg, Berlin, 1979).
M. A. Muratov and V. I. Chilin, Algebras of Measurable and Locally Measurable Operators (Pratsi In-tu Matem. NAN Ukraini, Kiev, 2007).
F. J. Yeadon, “Convergence ofMeasurable Operators,” Proc. Camb. Phil. Soc. 74, 257–268 (1973).
O. E. Tikhonov, “Continuity of Operator Functions in Topologies Connected to a Trace in Neumann’s Algebra,” Izv. Vyssh. Uchebn. Zaved., Mat. №1, 77–79 (1987) [Soviet Mathematics (Izv. VUZ) 31 (1), 75–77 (1987)].
D. A. Vladimirov, Boolean Algebras (Nauka, Moscow, 1969) [in Russian].
C. A. Akemann, T. Andersen, and G. K. Pedersen, “Triangle Inequalities in Operator Algebras,” Linear Multilinear Algebra 11(2), 167–178 (1982).
S. Sakai, C*-Algebras and W*-Algebras (Springer-Verlag, Berlin-Heidelberg-New York, 1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © B.S. Zakirov and V.I. Chilin, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 10, pp. 18–30.
About this article
Cite this article
Zakirov, B.S., Chilin, V.I. Noncommutative integration for traces with values in Kantorovich-Pinsker spaces. Russ Math. 54, 15–26 (2010). https://doi.org/10.3103/S1066369X10100026
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X10100026