Geometry of Banach limits and their applications

E. M. Semenov ; F. A. Sukochev ; A. S. Usachev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 4, pp. 725-763

Résumé

A Banach limit is a positive shift-invariant functional on $\ell_\infty$ which extends the functional $$ (x_1,x_2,\dots)\mapsto\lim_{n\to\infty}x_n $$ from the set of convergent sequences to $\ell_\infty$. The history of Banach limits has its origins in classical papers by Banach and Mazur. The set of Banach limits has interesting properties which are useful in applications. This survey describes the current state of the theory of Banach limits and of the areas in analysis where they have found applications. Bibliography: 137 titles.

Détail

Keywords: Banach limits, invariant Banach limits, almost convergent sequences, extreme points, dilation operator, Stone–Čech compactification, singular trace of an operator, non-commutative geometry.
Mots-clés : Cesàro operator

Bibliographie

[1] Z. U. Ahmad, Mursaleen, “An application of Banach limits”, Proc. Amer. Math. Soc., 103:1 (1988), 244–246 | DOI | MR | Zbl

[2] 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

[3] S. Albeverio, D. Guido, A. Ponosov, S. Scarlatti, “Singular traces and compact operators”, J. Funct. Anal., 137:2 (1996), 281–302 | DOI | MR | Zbl

[4] E. A. Alekhno, “Nekotorye spetsialnye svoistva funktsionalov Mazura. II”, Analiticheskie metody analiza i differentsialnykh uravnenii (AMADE-2006), Tr. 4-i Mezhdunar. konf. (Minsk, 2006), v. 2, BGU, IM NAN Belarusi, Minsk, 2006, 17–23

[5] 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

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

[7] 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

[8] E. Alekhno, 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

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

[10] E. Alekhno, E. Semenov, F. Sukochev, A. Usachev, “Invariant Banach limits and their extreme points”, Studia Math., 242:1 (2018), 79–107 | DOI | MR | Zbl

[11] C. D. Aliprantis, O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006, xx+376 pp. | DOI | MR | Zbl

[12] J. Appell, E. De Pascale, P. P. Zabrejko, “Some remarks on Banach limits”, Atti Sem. Mat. Fis. Univ. Modena, 42:1 (1994), 273–278 | MR | Zbl

[13] 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

[14] S. V. Astashkin, E. M. Semenov, “Lebesgue constants of the Walsh system and Banach limits”, Siberian Math. J., 57:3 (2016), 398–410 | DOI | DOI | MR | Zbl

[15] N. N. Avdeev, “On the space of almost convergent sequences”, Math. Notes, 105:3 (2019), 464–468 | DOI | DOI | MR | Zbl

[16] N. N. Avdeev, E. M. Semenov, A. S. Usachev, “Banach limits and a measure on the set of 0-1-sequences”, Math. Notes, 106:5 (2019), 833–836 | DOI | DOI | MR | Zbl

[17] S. Banach, Théorie des opérations linéaires, Monogr. Mat., 1, Inst. Mat. PAN, Warszawa, 1932, vii+254 pp. | MR | Zbl

[18] N. K. Bary, A treatise on trigonometric series, v. I, II, A Pergamon Press Book, The Macmillan Co., New York, 1964, xxiii+553 pp., xix+508 pp. | MR | MR | Zbl

[19] G. Bennett, N. J. Kalton, “Consistency theorems for almost convergence”, Trans. Amer. Math. Soc., 198 (1974), 23–43 | DOI | MR | Zbl

[20] M. M. Bogolyubov, “Pro deyaki ergodichni vlastivosti sutsilnikh grupp pretvoren”, Nauk. zap. Kiïv. derzh. un-tu im. T. G. Shevchenko. Fiz.-matem. zb., 4:3 (1939), 45–53

[21] É. Borel, “Les probabilités dénombrables et leurs applications arithmétiques”, Rend. Circ. Mat. Palermo, 27 (1909), 247–271 | DOI | Zbl

[22] R. E. Bruck, S. Reich, “Accretive operators, Banach limits and dual ergodic theorems”, Bull. Acad. Polon. Sci. Sér. Sci. Math., 29:11-12 (1981), 585–589 | MR | Zbl

[23] J. W. Calkin, “Abstract symmetric boundary conditions”, Trans. Amer. Math. Soc., 45:3 (1939), 369–442 | DOI | MR | Zbl

[24] A. L. Carey, J. Phillips, A. Rennie, F. Sukochev, “The Hochschild class of the Chern character for semifinite spectral triples”, J. Funct. Anal., 213:1 (2004), 111–153 | DOI | MR | Zbl

[25] A. Carey, J. Phillips, F. A. Sukochev, “Spectral flow and Dixmier traces”, Adv. Math., 173:1 (2003), 68–133 | DOI | MR | Zbl

[26] A. L. Carey, A. Rennie, A. Sedaev, F. Sukochev, “The {D}ixmier trace and asymptotics of zeta functions”, J. Funct. Anal., 249:2 (2007), 253–283 | DOI | MR | Zbl

[27] A. L. Carey, F. A. Sukochev, “Dixmier traces and some applications in non-commutative geometry”, Russian Math. Surveys, 61:6 (2006), 1039–1099 | DOI | DOI | MR | Zbl

[28] A. Carey, F. Sukochev, “Measurable operators and the asymptotics of heat kernels and zeta functions”, J. Funct. Anal., 262:10 (2012), 4582–4599 | DOI | MR | Zbl

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

[30] C. Chou, “Minimal sets and ergodic measures for $\beta N \setminus N$”, Illinois J. Math., 13:4 (1969), 777–788 | DOI | MR | Zbl

[31] A. Connes, “The action functional in non-commutative geometry”, Comm. Math. Phys., 117:4 (1988), 673–683 | DOI | MR | Zbl

[32] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp. | MR | Zbl

[33] A. Connes, E. McDonald, F. Sukochev, D. Zanin, “Conformal trace theorem for {J}ulia sets of quadratic polynomials”, Ergodic Theory Dynam. Systems, 39:9 (2019), 2481–2506 | DOI | MR | Zbl

[34] A. Connes, F. A. Sukochev, D. V. Zanin, “Trace theorem for quasi-Fuchsian groups”, Sb. Math., 208:10 (2017), 1473–1502 | DOI | DOI | MR | Zbl

[35] J. Connor, “Almost none of the sequences of 0's and 1's are almost convergent”, Internat. J. Math. Math. Sci., 13:4 (1990), 775–777 | DOI | MR | Zbl

[36] R. G. Cooke, “Generalizations of Banach–Hausdorff limits”, Proc. Amer. Math. Soc., 4:3 (1953), 410–417 | DOI | MR | Zbl

[37] Y. Cui, H. Hudzik, R. Pluciennik, “Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm”, Z. Anal. Anwendungen, 22:4 (2003), 789–817 | DOI | MR | Zbl

[38] G. Das, “Banach and other limits”, J. London Math. Soc. (2), 7:3 (1974), 501–507 | DOI | MR | Zbl

[39] G. Das, B. Kuttner, S. Nanda, “Some sequence spaces and absolute almost convergence”, Trans. Amer. Math. Soc., 283:2 (1984), 729–739 | DOI | MR | Zbl

[40] M. M. Day, “Amenable semigroups”, Illinois J. Math., 1:4 (1957), 509–544 | DOI | MR | Zbl

[41] M. M. Day, Normed linear spaces, Ergeb. Math. Grenzgeb. (N. F.), Springer-Verlag, Berlin–Göttingen–Heidelberg, 1958, iv+139 pp. | DOI | MR | MR | Zbl | Zbl

[42] M. M. Day, “Semigroups and amenability”, Semigroups (Wayne State Univ., Detroit, MI, 1968), Academic Press, New York, 1969, 5–53 | MR | Zbl

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

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

[45] P. Dodds, B. De Pagter, A. Sedaev, E. Semenov, F. Sukochev, “Singular symmetric functionals”, J. Math. Sci. (N. Y.), 124:2 (2004), 4867–4885 | DOI | MR | Zbl

[46] 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

[47] P. G. Dodds, B. de Pagter, E. M. Semenov, F. A. Sukochev, “Symmetric functionals and singular traces”, Positivity, 2:1 (1998), 47–75 | DOI | MR | Zbl

[48] R. G. Douglas, “On the measure-theoretic character of an invariant mean”, Proc. Amer. Math. Soc., 16 (1965), 30–36 | DOI | MR | Zbl

[49] N. Dunford, J. T. Schwartz, Linear operators, v. I, Pure Appl. Math., 7, General theory, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958, xiv+858 pp. | MR | MR | Zbl

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

[51] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York–Toronto–London, 1965, xiii+781 pp. | MR | Zbl | Zbl

[52] L. Fairchild, “Extreme invariant means without minimal support”, Trans. Amer. Math. Soc., 172 (1972), 83–93 | DOI | MR | Zbl

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

[54] J. Flores, F. L. Hernández, E. M. Semenov, P. Tradacete, “Strictly singular and power-compact operators on Banach lattices”, Israel J. Math., 188 (2012), 323–352 | DOI | MR | Zbl

[55] 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

[56] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge Univ. Press, London, 1974, xiv+266 pp. | MR | Zbl

[57] D. H. Fremlin, Well-distributed sequences and Banach density, Vers. of 28.3.11, 40 pp. https://www1.essex.ac.uk/maths/people/fremlin/n02j23.pdf

[58] 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

[59] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, 16, Van Nostrand Reinhold Co., New York–Toronto–London, 1969, ix+113 pp. | MR | Zbl | Zbl

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

[61] E. Hewitt, K. A. Ross, Abstract harmonic analysis, v. I, Grundlehren Math. Wiss., 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963, viii+519 pp. | MR | MR | Zbl

[62] M. Jerison, “The set of generalized limits of bounded sequences”, Canadian J. Math., 9 (1957), 79–89 | DOI | MR | Zbl

[63] V. M. Kadets, Kurs funktsionalnogo analiza, Khark. nats. un-t im. V. N. Karazina, Kharkov, 2006, 608 pp. | MR | Zbl

[64] N. Kalton, S. Lord, D. Potapov, F. Sukochev, “Traces of compact operators and the noncommutative residue”, Adv. Math., 235 (2013), 1–55 | DOI | MR | Zbl

[65] T. Kania, “Vector-valued invariant means revisited once again”, J. Math. Anal. Appl., 445:1 (2017), 797–802 | DOI | MR | Zbl

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

[67] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, xii+451 pp. | MR | MR | Zbl | Zbl

[68] G. Keller, L. C. Moore, Jr., “Invariant means on the group of integers”, Analysis and geometry, Bibliographisches Inst., Mannheim, 1992, 1–18 | MR | Zbl

[69] R. Kunisada, “Invariant linear functionals on $L^\infty(R_+)$”, J. Math. Anal. Appl., 481:1 (2020), 123452, 22 pp. | DOI | MR | Zbl

[70] J. C. Kurtz, “Almost convergent vector sequences”, Tohoku Math. J. (2), 22:4 (1970), 493–498 | DOI | MR | Zbl

[71] J. C. Kurtz, “Almost convergence in Banach spaces”, Tohoku Math. J. (2), 24:3 (1972), 389–399 | DOI | MR | Zbl

[72] S. S. Kutateladze, Fundamentals of functional analysis, Kluwer Texts Math. Sci., 12, Kluwer Acad. Publ., Dordrecht, 1996, xiv+276 pp. | DOI | MR | MR | Zbl | Zbl

[73] S. Li, C. Li, Y.-C. Li, “On $\sigma$-limit and $s\sigma$-limit in Banach spaces”, Taiwanese J. Math., 9:3 (2005), 359–371 | DOI | MR | Zbl

[74] X. Li, W. Shen, C. 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

[75] Y.-C. Li, “Almost convergence of sequences in Banach spaces in weak, strong and absolute senses”, Taiwanese J. Math., 10:1 (2006), 209–218 | DOI | MR | Zbl

[76] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Ergeb. Math. Grenzgeb., II, Function spaces, Springer-Verlag, Berlin–New York, 1979, x+243 pp. | MR | Zbl

[77] 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

[78] S. Lord, F. A. Sukochev, D. Zanin, “Advances in Dixmier traces and applications”, Advances in noncommutative geometry, Springer, Cham, 2020, 491–584 | DOI

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

[80] W. A. J. Luxemburg, “Nonstandard hulls, generalized limits and almost convergence”, Analysis and geometry. Trends in research and teaching, eds. B. Fuchssteiner, W. A. J. Luxemburg, B. I. Wissenschaftsverlag, Mannheim, 1992, 19–45 | MR | Zbl

[81] W. A. J. Luxemburg, B. de Pagter, “Invariant means for positive operators and semigroups”, Circumspice. Various papers in and around mathematics in honor of Arnoud van Rooij, Catholic Univ. of Nijmegen, Nijmegen, 2001, 31–55

[82] 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

[83] S. Mazur, “O metodach sumowalności”, Ann. Soc. Polon. Math. (Suppl.), 1929, 102–107

[84] S. Mazur, “On the generalized limit of bounded sequences”, Colloq. Math., 2 (1951), 173–175 | DOI | MR | Zbl

[85] J. Mercer, “Sturm–Liouville series of normal functions in the theory of integral equations”, Philos. Trans. R. Soc. Lond. Ser. A, 211 (1912), 111–198 | DOI | Zbl

[86] V. D. Milman, “Operatory klassa $c_0$ i $c_0^*$”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, 10, Izd-vo Khark. un-ta, Kh., 1970, 15–26 | MR | Zbl

[87] F. Móricz, B. E. Rhoades, “Almost convergence of double sequences and strong regularity of summability matrices”, Math. Proc. Cambridge Philos. Soc., 104:2 (1988), 283–294 | DOI | MR | Zbl

[88] Mursaleen, “On some new invariant matrix methods of summability”, Quart. J. Math. Oxford Ser. (2), 34 (1983), 77–86 | DOI | MR | Zbl

[89] Mursaleen, A. K. Gaur, T. A. Chishti, “On some new sequence spaces of invariant means”, Acta Math. Hungar., 75:3 (1997), 209–214 | DOI | MR | Zbl

[90] 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

[91] S. Nanda, “Strongly almost summable and strongly almost convergent sequences”, Acta Math. Hungar., 49:1-2 (1987), 71–76 | DOI | MR | Zbl

[92] J. v. Neumann, “Zur allgemeinen Theorie des Masses”, Fund. Math., 13 (1929), 73–116 | DOI | Zbl

[93] R. Nillsen, “Nets of extreme Banach limits”, Proc. Amer. Math. Soc., 55:2 (1976), 347–352 | DOI | MR | Zbl

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

[95] A. Pietsch, “Einige neue Klassen von kompakten linearen Abbildungen”, Rev. Math. Pures Appl., 8 (1963), 427–447 | MR | Zbl

[96] A. Pietsch, “Traces and shift invariant functionals”, Math. Nachr., 145 (1990), 7–43 | DOI | MR | Zbl

[97] A. Pietsch, “Connes–Dixmier versus Dixmier traces”, Integral Equations Operator Theory, 77:2 (2013), 243–259 | DOI | MR | Zbl

[98] A. Pietsch, “Traces of operators and their history”, Acta Comment. Univ. Tartu. Math., 18:1 (2014), 51–64 | DOI | MR | Zbl

[99] A. Pietsch, “Traces on operator ideals and related linear forms on sequence ideals (part I)”, Indag. Math. (N. S.), 25:2 (2014), 341–365 | DOI | MR | Zbl

[100] A. Pietsch, “Traces on operator ideals and related linear forms on sequence ideals (part II)”, Integral Equations Operator Theory, 79:2 (2014), 255–299 | DOI | MR | Zbl

[101] A. Pietsch, “Traces on operator ideals and related linear forms on sequence ideals (part III)”, J. Math. Anal. Appl., 421:2 (2015), 971–981 | DOI | MR | Zbl

[102] A. Pietsch, “A new approach to operator ideals on {H}ilbert space and their traces”, Integral Equations Operator Theory, 89:4 (2017), 595–606 | DOI | MR | Zbl

[103] A. Pietsch, “The spectrum of shift operators and the existence of traces”, Integral Equations Operator Theory, 90:2 (2018), 17, 13 pp. | DOI | MR | Zbl

[104] W. Prager, J. Schwaiger, “Vector valued Banach limits and generalizations applied to the inhomogeneous Cauchy equation”, Aequationes Math., 93:1 (2019), 257–275 | DOI | MR | Zbl

[105] R. A. Raimi, “Invariant means and invariant matrix methods of summability”, Duke Math. J., 30 (1963), 81–94 | DOI | MR | Zbl

[106] R. A. Raimi, “Factorization of summability-preserving generalized limits”, J. London Math. Soc. (2), 22:3 (1980), 398–402 | DOI | MR | Zbl

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

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

[109] A. A. Sedaev, “Ob approksimatsii predelov Banakha elementami prostranstva $\ell_1$”, Vestn. Voronezh. gos. un-ta. Ser. Fiz. Matem., 2006, no. 1, 187–192 | Zbl

[110] A. A. Sedaev, F. A. Sukochev, “Dixmier measurability in Marcinkiewicz spaces and applications”, J. Funct. Anal., 265:12 (2013), 3053–3066 | DOI | MR | Zbl

[111] E. M. Semenov, A. M. Shteĭnberg, “Norm estimates of operator blocks in Banach lattices”, Math. USSR-Sb., 54:2 (1986), 317–333 | DOI | MR | Zbl

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

[113] E. M. Semenov, F. A. Sukochev, “Characteristic functions of Banach limits”, Siberian Math. J., 51:4 (2010), 723–727 | DOI | MR | Zbl

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

[115] E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Structure properties of the Banach limit set”, Dokl. Math., 84:3 (2011), 802–803 | DOI | MR | Zbl

[116] 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

[117] E. M. Semenov, F. A. Sukochev, A. S. Usachev, “The main classes of invariant Banach limits”, Izv. Math., 83:1 (2019), 124–150 | DOI | DOI

[118] 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

[119] E. Semenov, F. Sukochev, A. Usachev, D. Zanin, “Dilation invariant Banach limits”, Indag. Math. (to appear) ; 2019, 10 pp., arXiv: 1910.10899 | DOI

[120] E. Semenov, F. Sukochev, A. Usachev, D. Zanin, “Invariant Banach limits and applications to noncommutative geometry”, Pacific Math. J., 306:1 (2020), 357–373 | DOI | MR | Zbl

[121] E. M. Semenov, A. S. Usachev, “Fourier–Haar coefficients and Banach limits”, Dokl. Math., 79:2 (2009), 191–192 | DOI | MR | Zbl

[122] E. M. Semenov, A. S. Usachev, O. O. Khorpyakov, “The space of almost convergent sequences”, Dokl. Math., 74:1 (2006), 587–588 | DOI | MR | Zbl

[123] R. Sikorski, “On the existence of the generalized limit”, Studia Math., 12 (1951), 117–124 | DOI | MR | Zbl

[124] M. A. Sofi, Banach limits – some new thoughts and perspectives, 2019, 16 pp., arXiv: 1906.04168

[125] L. Sucheston, “On existence of finite invariant measures”, Math. Z., 86 (1964), 327–336 | DOI | MR | Zbl

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

[127] 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

[128] F. Sukochev, A. Usachev, D. Zanin, “On the distinction between the classes of Dixmier and Connes–Dixmier traces”, Proc. Amer. Math. Soc., 141:6 (2013), 2169–2179 | DOI | MR | Zbl

[129] F. Sukochev, A. Usachev, D. Zanin, “Dixmier traces generated by exponentiation invariant generalised limits”, J. Noncommut. Geom., 8:2 (2014), 321–336 | DOI | MR | Zbl

[130] F. Sukochev, A. Usachev, D. Zanin, “Singular traces and residues of the $\zeta$-function”, Indiana Univ. Math. J., 66:4 (2017), 1107–1144 | DOI | MR | Zbl

[131] F. Sukochev, D. Zanin, “$\zeta$-function and heat kernel formulae”, J. Funct. Anal., 260:8 (2011), 2451–2482 | DOI | MR | Zbl

[132] F. Sukochev, D. Zanin, “Which traces are spectral?”, Adv. Math., 252 (2014), 406–428 | DOI | MR | Zbl

[133] M. Talagrand, “Géométrie des simplexes de moyennes invariantes”, J. Funct. Anal., 34:2 (1979), 304–337 | DOI | MR | Zbl

[134] R. Tanaka, “On vector-valued Banach limits with values in $\mathcal B(\mathcal H)$”, Publ. Math. Debrecen, 92:3-4 (2018), 471–480 | DOI | MR | Zbl

[135] A. S. Usachev, “Transformations in the space of almost convergent sequences”, Siberian Math. J., 49:6 (2008), 1136–1137 | DOI | MR | Zbl

[136] A. Usachev, “On Fourier–Haar coefficients of the function from the Marcinkiewicz space”, J. Math. Anal. Appl., 414:1 (2014), 110–124 | DOI | MR | Zbl

[137] S. Wagon, The Banach–Tarski paradox, Corr. reprint of the 1985 original, Cambridge Univ. Press, Cambridge, 1994, xviii+253 pp. | MR | Zbl