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Order and geometric properties of the set of Banach limits. (English. Russian original) Zbl 1373.46008

St. Petersbg. Math. J. 28, No. 3, 299-321 (2017); translation from Algebra Anal. 28 No. 3, 3-35 (2016).
Denote by \(\mathfrak{B}\subseteq \ell_{\infty}^*\) the set of all Banach limits. The authors prove various results concerning the structure of the set \(\mathfrak{B}\) and the set \(\text{ext}(\mathfrak{B})\) of its extreme points. For example, they prove that there is an element \(B\in \text{ext}(\mathfrak{B})\) such that \(Bx=0\) for every \(x=(x_n)\in \ell_{\infty}\) for which \((|x_n|)\) is Cesàro-convergent to \(0\), and that the set \(\mathfrak{B}\) does not have the fixed point property for affine, nonexpansive, weak\(^*\)-sequentially continuous mappings.
They also show that the so-called stabilizer \(\mathcal{D}(\text{ac}_0)\) and the ideal stabilizer \(\mathcal{I}(\text{ac}_0)\) of the space \(\text{ac}_0\) of all sequences which are almost convergent to \(0\) (i.e., \(Bx=0\) for every \(B\in \mathfrak{B}\)) are not complemented in \(\ell_{\infty}\).
The authors also study the sets \(\mathfrak{B}(\sigma_m)\) of all Banach limits which are invariant under the \(m\)-th dilation operator \(\sigma_m\). For instance, it is shown that \(\|B_1-B_2\|=2\) for every \(B_1\in \mathfrak{B}(\sigma_m)\) and every \(B_2\in \text{ext}(\mathfrak{B})\).

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
40H05 Functional analytic methods in summability
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[1] Al3 E. A. Alekhno, Some special properties of Mazur’s functionals. II, Proc. Internat. Conf. AMADE-2006, vol. 2, Minsk, 2006, pp. 17-23.
[2] Al \bysame , Superposition operator on the space of sequences almost converging to zero, Cent. Eur. J. Math. 10 (2012), no. 2, 619-645. · Zbl 1256.47036
[3] Al2 \bysame , On Banach-Mazur limits, Indag. Math. (N.S.) 26 (2015), no. 4, 581-614. · Zbl 1337.46016
[4] ASSU1 E. A. Alekhno, E. M. Semenov, F. A. Sukochev, and A. S. Usachev, Order properties of a set of Banach limits, Dokl. Akad. Nauk 460 (2015), no. 2, 131-132; English transl., Dokl. Math. 91 (2015), no. 1, 20-22. · Zbl 1333.46011
[5] AlB C. D. Aliprantis and K. C. Border, Infinite dimensional analysis. A hitchhiker’s guide, Springer-Verlag, Berlin, 2006. · Zbl 1156.46001
[6] AlBurk2 C. D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Math. Surveys Monogr., vol. 105, Amer. Math. Soc., Providence, RI, 2003. · Zbl 1043.46003
[7] AlBurk \bysame , Positive operators, Pure Appl. Math., vol. 119, Acad. Press, Orlando, FL, 1985. · Zbl 0608.47039
[8] B S. Banach, Th\'eorie des operations lin\'eaires, Chelsea Publ. Co., New York, 1955. · Zbl 0067.08902
[9] Carothers N. L. Carothers, A short course on Banach space theory, London Math. Soc. Student Text., vol. 64, Cambridge Univ. Press, Cambridge, 2005. · Zbl 1072.46001
[10] CC C. Chou, On the size of the set of left invariant means on a semigroup, Proc. Amer. Math. Soc. 23 (1969), no. 1, 199-205. · Zbl 0188.19006
[11] CHPlu Y. Cui, H. Hudzik, and R. Pluchenik, Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm, Z. Anal. Anwendungen 22 (2003), no. 4, 789-817. · Zbl 1060.46020
[12] DPSSS2 P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, and F. A. Sukochev, Singular symmetric functionals and Banach limits with additional invariance properties, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 6, 111-136; English transl., Izv. Math. 67 (2003), no. 6, 1187-1212. · Zbl 1075.46028
[13] Eberlein W. F. Eberlein, Banach-Hausdorf limits, Proc. Amer. Math. Soc. 1 (1950), no. 5, 662-665. · Zbl 0039.12102
[14] Jerison M. Jerison, The set of all generalized limits of bounded sequences, Canad. J. Math. 9 (1957), no. 1, 79-89. · Zbl 0077.31004
[15] KM G. Keller, L. C. Moore, Jr., Invariant means on the group of integers, Analysis and Geometry, Bibliographisches Inst., Mannheim, 1992, pp. 1-18. · Zbl 0786.43002
[16] L G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190. · Zbl 0031.29501
[17] Luxemburg W. A. J. Luxemburg, Nonstandard hulls, generalized limits and almost convergence, Analysis and Geometry, Bibliographisches Inst., Mannheim, 1992, pp. 19-45. · Zbl 0765.26012
[18] Nillsen R. Nillsen, Nets of extreme Banach limits, Proc. Amer. Math. Soc. 55 (1976), no. 2, 347-352. · Zbl 0302.28014
[19] Rickart_book Ch. E. Rickart, General theory of Banach algebras, Univ. Ser. Higher Math., D. Van Nostrand Co., Princeton, N.J., 1974.
[20] SS_JFA E. M. Semenov and F. A. Sukochev, Invariant Banach limits and applications, J. Funct. Anal. 256 (2010), no. 6, 1517-1541. · Zbl 1205.46012
[21] SS_pos \bysame , Extreme points of the set of Banach limits, Positivity 17 (2013), no. 1, 163-170. · Zbl 1287.46014
[22] SSU2 E. M. Semenov, F. A. Sukochev, and A. S. Usachev, Geometric properties of the set of Banach limits, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), no. 3, 177-204; English transl., Izv. Math. 78 (2014), no. 3, 596-620. · Zbl 1309.40003
[23] SSUZ E. M. Semenov, F. A. Sukochev, A. S. Usachev, and D. V. Zanin, Banach limits and traces on \(L_1,\infty \), Actv. Math. 285 (2015), 568-628. · Zbl 1405.47007
[24] S L. Sucheston, Banach limits, Amer. Math. Monthly 74 (1967), no. 3, 308-311. · Zbl 0148.12202
[25] Talagrand M. Talagrand, Geometrie des simplexes de moyennes invariantes, J. Funct. Anal. 34 (1979), no. 2, 304-337. · Zbl 0431.43001
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