Abstract
We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.
Similar content being viewed by others
References
S. Albeverio, D. Guido, A. Ponosov and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal. 137(1996), 281-302.
M. Braverman and A. Mekler, The Hardy-Littlewood property for symmetric spaces, Sib. Mat. Zhur. 18(1977) 522-540 (Russian).
C. Bennett and R. Sharpley, Interpolation of Operators,Academic Press, 1988.
A. Connes, Noncommutative Geometry, Academic Press, 1994.
V. I. Chilin and F. A. Sukochev, Symmetric spaces over semifinite von Neumann algebras, Soviet Math. Dokl. 42(1991), 97-101.
V. I. Chilin and F. A. Sukochev, Weak convergence in non-commutative symmetric spaces, J. Operator Theory 31(1994), 35-65.
J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris 262(1966) A1107-A1108.
P. G. Dodds, T. K. Dodds and B. de Pagter, Non-commutative Banach function spaces, Math. Z. 201(1989), 583-597.
P. G. Dodds, T. K. Dodds and B. de Pagter, Fully symmetric operator spaces, Integr. Equat. Oper. Th. 15(1992), 942-972.
T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123(1986), 269-300.
I. C. Gohberg and M. G. Krein, Introduction to the theory of non-selfadjoint operatorsTranslations of Mathematical Monographs, vol.18, AMS (1969).
D. Guido and T. Isola, Singular traces on semifinite von Neumann algebras, J. Funct. Anal. 134(1995), 451-485.
A. Kaminska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147(1990) 29-38.
S. G. Krein, Ju. I. Petunin and E. M. Semenov, Interpolation of linear operators, Translations of Mathematical Monographs, Amer. Math. Soc. 54(1982).
G. Ya. Lozanovskii, On localizable functionals in vector lattices,in: Theory of Functions, Functional Analysis and their Applications, 19(1974) 66-80 (Russian).
G. Ya. Lozanovskii, On the representation of linear functionals in Marcinkiewicz spaces,Izvestiya VUZov. Matematika N1 (188) (1978) 43–53. (Russian)
G. Ya. Lozanovskii, A supplement to the paper “On localizable functionals in vector lattices”, J. Soviet Math. 14(1980), 1170-1173.
E. Nelson, Notes on non-commutative integration, J. Functional Anal. 15(1974), 103-116.
R. Prinzis, Traces Residuelles et Asymptotique du Spectre d'Operateurs Pseudo-Differentiels, Ph. D. Thesis, Lyon, 1995.
G. Russu, Symmetric spaces not having the majorant property, Mat. Issled. 4(1969), 82-93 (Russian).
A. A. Sedaev, A symmetric space which is intermediate for the couple {L 1 ;L ∞} but which is not a subspace of any interpolation space, in Studies in the theory of functions of many real variables, Collection of papers, Yaroslavl (1990), (134-139).
K. Sundaresan, Additive functionals on Orlicz spaces, Studia Math. 32(1969), 269-276.
M. Terp, L p -spaces associated with von Neumann algebras,Notes, Copenhagen Univ. (1981).
J. V. Varga, Traces on irregular ideals, Proc. Amer. Math. Soc. 107(1989) 715-723.
A. C. Zaanen, Riesz Spaces II, (North-Holland), Amsterdam-New York-Oxford (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dodds, P., de Pagter, B., Semenov, E. et al. Symmetric Functionals and Singular Traces. Positivity 2, 47–75 (1998). https://doi.org/10.1023/A:1009720826217
Issue Date:
DOI: https://doi.org/10.1023/A:1009720826217