Geodesic
Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 90-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider Marcinkiewicz spaces of functions measurable on a semiaxis that admit a wide set of singular symmetric Dixmier functionals. For elements of these spaces we study the measurability property introduced by A. Connes. We establish that this property is closely connected with the Tauberian property (which is more strong) but is not reduced to it. We specify the maximal subspace of the Marcinkiewicz space such that for its elements both properties are equivalent. We prove that this subspace is not reducible to other known subspaces of the Marcinkiewicz space and that it plays an important role in the theory of Dixmier functionals.
Keywords: singular Dixmier traces, singular symmetric functionals, Connes measurable elements. 
@article{IVM_2009_12_a11,
     author = {A. A. Sedaev},
     title = {Singular symmetric functionals and stabilizing subspaces of {Marcinkiewicz} spaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {90--94},
     publisher = {mathdoc},
     number = {12},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_12_a11/}
}
TY  - JOUR
AU  - A. A. Sedaev
TI  - Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2009
SP  - 90
EP  - 94
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2009_12_a11/
LA  - ru
ID  - IVM_2009_12_a11
ER  - 
%0 Journal Article
%A A. A. Sedaev
%T Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2009
%P 90-94
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2009_12_a11/
%G ru
%F IVM_2009_12_a11
A. A. Sedaev. Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2009), pp. 90-94. http://geodesic.mathdoc.fr/item/IVM_2009_12_a11/

[1] Dixmier J., “Existence de traces non normales”, C. R. Acad. Sci. Paris, 262 (1966), A1107–A1108 | MR

[2] Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994 | MR | Zbl

[3] Carey A. L., Phillips J., Sukochev F. A., “Spectral flow and Dixmier traces”, Adv. Math., 173:1 (2003), 68–113 | DOI | MR | Zbl

[4] Lord S., Sedaev A., Sukochev F., “Dixmier traces as singular symmetric functionals and applications to measurable operators”, J. Funct. Anal., 224 (2005), 72–206 | MR

[5] Azamov N. A., Sukochev F. A., “A Lidskii type formula for Dixmier traces”, C. R. Math. Acad. Sci. Paris, 340:2 (2005), 107–112 | MR | Zbl

[6] Dodds P., de Pagter B., Semenov E., Sukochev F., “Symmetric functionals and singular traces”, Positivity, 2 (1998), 47–75 | DOI | MR | Zbl

[7] Dodds P. G., de Pagter B., Sedaev A. A., Semenov E. M., Sukochev F. A., “Singulyarnye simmetrichnye funktsionaly”, Zap. nauch. semin. POMI, 290, 2002, 42–71 | MR | Zbl

[8] Dodds P. G., de Pagter B., Sedaev A. A., Semenov E. M., Sukochev F. A., “Singulyarnye simmetrichnye funktsionaly i banakhovy predely s dopolnitelnymi svoistvami invariantnosti”, Izv. RAN. Ser. matem., 67:6 (2003), 111–136 | MR | Zbl

[9] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978, 400 pp. | MR

[10] Keri A. L., Sukochev F. A., “Sledy Diksme i nekotorye prilozheniya v nekommutativnoi geometrii”, UMN, 61:6 (2006), 45–110 | MR | Zbl

[11] Khardi G., Raskhodyaschiesya ryady, Faktorial Press, M., 2006, 504 pp.