Geodesic
The main classes of invariant Banach limits
Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 124-150.

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

We study functional characteristics and extreme points of the set of Banach limits on the space of bounded sequences. We also consider Banach limits with additional invariance properties with respect to the dilation operator and the Cesàro operator.
Keywords: invariant Banach limits, extreme points, Cesàro operator, dilation operator. 
@article{IM2_2019_83_1_a5,
     author = {E. M. Semenov and F. A. Sukochev and A. Usachev},
     title = {The main classes of invariant {Banach} limits},
     journal = {Izvestiya. Mathematics },
     pages = {124--150},
     publisher = {mathdoc},
     volume = {83},
     number = {1},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a5/}
}
TY  - JOUR
AU  - E. M. Semenov
AU  - F. A. Sukochev
AU  - A. Usachev
TI  - The main classes of invariant Banach limits
JO  - Izvestiya. Mathematics 
PY  - 2019
SP  - 124
EP  - 150
VL  - 83
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a5/
LA  - en
ID  - IM2_2019_83_1_a5
ER  - 
%0 Journal Article
%A E. M. Semenov
%A F. A. Sukochev
%A A. Usachev
%T The main classes of invariant Banach limits
%J Izvestiya. Mathematics 
%D 2019
%P 124-150
%V 83
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a5/
%G en
%F IM2_2019_83_1_a5
E. M. Semenov; F. A. Sukochev; A. Usachev. The main classes of invariant Banach limits. Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 124-150. http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a5/

[1] S. Banach, Théorie des opérations linéaires, Monografie Matematyczne, 1, Instytut Matematyczny PAN, Warszawa, 1932, vii+254 pp. | MR | Zbl

[2] A. Guichardet, “La trace de Dixmier et autres traces”, Enseign. Math., 61:3-4 (2015), 461–481 | DOI | MR | Zbl

[3] J. Dixmier, “Existence de traces non normales”, C. R. Acad. Sci. Paris Sér. A-B, 262 (1966), A1107–A1108 | MR | Zbl

[4] M. K. Roychowdhury, “Quantization dimension for Gibbs-like measures on cookie-cutter sets”, Kyoto J. Math., 54:2 (2014), 239–257 | DOI | MR | Zbl

[5] Y. Peres, “Application of Banach limits to the study of sets of integers”, Israel J. Math., 62:1 (1988), 17–31 | DOI | MR | Zbl

[6] R. Nillsen, “A characterisation of ergodic measures”, J. Austral. Math. Soc., 19:2 (1975), 222–224 | DOI | MR | Zbl

[7] D. H. Fremlin, M. Talagrand, “A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means”, Math. Z., 168:2 (1979), 117–142 | DOI | MR | Zbl

[8] Xin Li, Wenxian Shen, Chunyou Sun, “Invariant measures for complex-valued dissipative dynamical systems and applications”, Discrete Contin. Dyn. Syst. Ser. B, 22:6 (2017), 2427–2446 | DOI | MR | Zbl

[9] L. Rotem, “Banach limit in convexity and geometric means for convex bodies”, Electron. Res. Announc. Math. Sci., 23 (2016), 41–51 | DOI | MR

[10] K. Matomäki, M. Radziwiłł, T. Tao, “Sign patterns of the Liouville and Möbius functions”, Forum Math. Sigma, 4 (2016), 14, 44 pp. | DOI | MR | Zbl

[11] C. Foias, R. M. S. Rosa, R. M. Temam, “Convergence of time averages of weak solutions of the three-dimensional Navier–Stokes equations”, J. Stat. Phys., 160:3 (2015), 519–531 | DOI | MR | Zbl

[12] E. Semenov, F. Sukochev, A. Usachev, D. Zanin, “Banach limits and traces on $\mathscr L_{1,\infty}$”, Adv. Math., 285 (2015), 568–628 | DOI | MR | Zbl

[13] J. B. Deeds, “Summability of vector sequences”, Studia Math., 30 (1968), 361–372 | DOI | MR | Zbl

[14] A. Aizpuru, R. Armario, F. J. García-Pacheco, F. J. Pérez-Fernández, “Banach limits and uniform almost summability”, J. Math. Anal. Appl., 379:1 (2011), 82–90 | DOI | MR | Zbl

[15] R. Armario, F. J. García-Pacheco, F. J. Pérez-Fernández, “On vector-valued Banach limits”, Funct. Anal. Appl., 47:4 (2013), 315–318 | DOI | DOI | MR | Zbl

[16] G. Fichtenholz, L. Kantorovitch, “Sur les opérations linéaires dans l'espace des fonctions bornées”, Studia Math., 5 (1934), 69–98 | DOI | Zbl

[17] M. Nakamura, S. Kakutani, “Banach limits and the Čech compactification of a countable discrete set”, Proc. Imp. Acad. Tokyo, 19:5 (1943), 224–229 | DOI | MR | Zbl

[18] Ching Chou, “On the size of the set of left invariant means on a semigroup”, Proc. Amer. Math. Soc., 23:1 (1969), 199–205 | DOI | MR | Zbl

[19] E. Alekno, E. Semenov, F. Sukochev, A. Usachev, “On the structure of invariant Banach limits”, C. R. Math. Acad. Sci. Paris, 354:12 (2016), 1195–1199 | DOI | MR | Zbl

[20] E. A. Alekno, E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Banach limits: invariance and functional characteristics”, Dokl. Math., 96:1 (2017), 305–307 | DOI | MR | Zbl

[21] E. A. Alekhno, “Superposition operator on the space of sequences almost converging to zero”, Cent. Eur. J. Math., 10:2 (2012), 619–645 | DOI | MR | Zbl

[22] E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Geometric properties of the set of Banach limits”, Izv. Math., 78:3 (2014), 596–620 | DOI | DOI | MR | Zbl

[23] G. G. Lorentz, “A contribution to the theory of divergent sequences”, Acta Math., 80 (1948), 167–190 | DOI | MR | Zbl

[24] L. Sucheston, “Banach limits”, Amer. Math. Monthly, 74:3 (1967), 308–311 | DOI | MR | Zbl

[25] W. F. Eberlein, “Banach–Hausdorff limits”, Proc. Amer. Math. Soc., 1 (1950), 662–665 | DOI | MR | Zbl

[26] P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, F. A. Sukochev, “Singular symmetric functionals and Banach limits with additional invariance properties”, Izv. Math., 67:6 (2003), 1187–1212 | DOI | DOI | MR | Zbl

[27] E. M. Semenov, F. A. Sukochev, “Invariant Banach limits and applications”, J. Funct. Anal., 256:6 (2010), 1517–1541 | DOI | MR | Zbl

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

[29] E. Semenov, F. Sukochev, “Extreme points of the set of Banach limits”, Positivity, 17:1 (2013), 163–170 | DOI | MR | Zbl

[30] L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press, Oxford–Elmsford, N.Y., 1982, xiv+589 pp. | MR | MR | Zbl | Zbl

[31] Yu. S. Ochan, Sbornik zadach i teorem po teorii funktsii deistvitelnogo peremennogo, Prosveschenie, M., 1965, 228 pp.

[32] E. Alekhno, E. Semenov, F. Sukochev, A. Usachev, “Order and geometric properties of the set of Banach limits”, St. Petersburg Math. J., 28:3 (2017), 299–321 | DOI | MR | Zbl

[33] E. A. Alekhno, “On Banach–Mazur limits”, Indag. Math. (N.S.), 26:4 (2015), 581–614 | DOI | MR | Zbl

[34] S. Lord, F. Sukochev, D. Zanin, Singular traces. Theory and applications, De Gruyter Stud. Math., 46, De Gruyter, Berlin, 2013, xvi+452 pp. | MR | Zbl

[35] F. Sukochev, A. Usachev, D. Zanin, “Generalized limits with additional invariance properties and their applications to noncommutative geometry”, Adv. Math., 239 (2013), 164–189 | DOI | MR | Zbl