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EQUIVARIANT GEOMETRY OF BANACH SPACES AND TOPOLOGICAL GROUPS

Part of: Normed linear spaces and Banach spaces; Banach lattices Special aspects of infinite or finite groups

Published online by Cambridge University Press:  07 September 2017

CHRISTIAN ROSENDAL
CHRISTIAN ROSENDAL*
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA; rosendal.math@gmail.com

Abstract

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We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, that is, continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

MSC classification

Primary: 46B80: Nonlinear classification of Banach spaces; nonlinear quotients 46B20: Geometry and structure of normed linear spaces
Secondary: 20F65: Geometric group theory

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e22
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

References

Aharoni, I., ‘Every separable metric space is Lipschitz equivalent to a subset of c 0 ’, Israel J. Math. 19 (1974), 284–291.Google Scholar
Aharoni, I., ‘Uniform embeddings of Banach spaces’, Israel J. Math. 27 (1977), 174–179.Google Scholar
Aharoni, I., Maurey, B. and Mityagin, B. S., ‘Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces’, Israel J. Math. 52(3) (1985), 251–265.Google Scholar
Alaoglu, L. and Birkhoff, G., ‘General ergodic theorems’, Ann. of Math. (2) 41 (1940), 293–309.Google Scholar
Albiac, F. and Kalton, N., Topics in Banach Space Theory (Springer, New York, 2006).Google Scholar
Bader, U., Gelander, T. and Monod, N., ‘A fixed point theorem for L 1 ’, Invent. Math. 189(1) (2012), 143–148.Google Scholar
Bader, U., Rosendal, C. and Sauer, R., ‘On the cohomology of weakly almost periodic group representations’, J. Topol. Anal. 6(2) (2014), 153–165.Google Scholar
Bekka, M. E. B., Chérix, P.-A. and Valette, A., ‘Proper affine isometric actions of amenable groups’, inNovikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach 1993), London Mathematical Society Lecture Note Series, 227 (Cambridge University Press, Cambridge, 1995), 1–4.Google Scholar
Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s Property (T), New Mathematical Monographs, 11 (Cambridge University Press, Cambridge, 2008).Google Scholar
Ben Yaacov, I., Berenstein, A. and Ferri, S., ‘Reflexive representability and stable metrics’, Math. Z. 267 (2011), 129–138.Google Scholar
Benyamini, I. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. 1, (American Mathematical Society, Providence, Rhode Island, 2000).Google Scholar
Braga, B., ‘On weaker notions of nonlinear embeddings between Banach spaces’, Preprint, 2016.Google Scholar
Braga, B., ‘Coarse and uniform embeddings’, J. Funct. Anal. 272(5) (2017), 1852–1875.Google Scholar
Braga, B., ‘Asymptotic structure and coarse Lipschitz geometry of Banach spaces’, Stud. Math. 237(1) (2017), 71–97.Google Scholar
Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J.-L., ‘Fonctions de type positif sur les espaces L p ’, C. R. Acad. Sci. Paris 261 (1965), 2153–2156. (French).Google Scholar
Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J.-L., ‘Lois stables et espaces L p ’, Ann. Inst. H. Poincaré Sect. B (N.S.) 2 (1965/1966), 231–259. (French).Google Scholar
Brown, N. and Guentner, E., ‘Uniform embedding of bounded geometry spoaces into reflexive Banach space’, Proc. Amer. Math. Soc. 133(7) (2005), 2045–2050.Google Scholar
Chérix, P.-A., Cowling, M., Jolissant, P., Julg, P. and Valette, A., Groups with the Haagerup Property: Gromov’s a-T-Menability, (Basel, Birkhäuser, 2001).Google Scholar
Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40 (1936), 396–414.Google Scholar
de Cornulier, Y., Tessera, R. and Valette, A., ‘Isometric group actions on Hilbert spaces: growth of cocycles’, GAFA, Geom. Funct. Anal. 17 (2007), 770–792.Google Scholar
Davis, W. J., Figiel, T., Johnson, W. B. and Pełczyński, A., ‘Factoring weakly compact operators’, J. Funct. Anal. 17 (1974), 311–327.Google Scholar
Enflo, P., ‘Banach spaces which can be given an equivalent uniformly convex norm’, Israel J. Math. 13 (1972), 281–288.Google Scholar
Fabian, M., Habala, P., Hájek, P., Montesinos, V. and Zizler, V., ‘Banach space theory. The basis for linear and nonlinear analysis’, inCMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, New York, 2011).Google Scholar
Ferenczi, V., ‘A uniformly convex hereditarily indecomposable Banach space’, Israel J. Math. 102 (1997), 199–225.Google Scholar
Gervirtz, J., ‘Stability of isometries on Banach spaces’, Proc. Amer. Math. Soc. 89 (1983), 633–636.Google Scholar
Grothendieck, A., ‘Critères de compacité dans les espaces fonctionnels généraux’, Amer. J. Math. 74 (1952), 168–186.Google Scholar
Gruber, P. M., ‘Stability of isometries’, Trans. Amer. Math. Soc. 245 (1978), 263–277.Google Scholar
Guentner, E. and Kaminker, J., ‘Exactness and uniform embeddability of discrete groups’, J. Lond. Math. Soc. (2) 70 (2004), 703–718.Google Scholar
Haagerup, U., ‘An example of a nonnuclear C ∗ -algebra, which has the metric approximation property’, Invent. Math. 50 (1978), 279–293.Google Scholar
Haagerup, U. and Przybyszewska, A., ‘Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces’, Preprint, 2006, arXiv:0606794.Google Scholar
de la Harpe, P., ‘Moyennabilité du groupe unitaire et propriété P de Schwartz des algèbres de von Neumann’, inAlgèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978), Lecture Notes in Mathematics, 725 (Springer, Berlin, 1979), 220–227.Google Scholar
Herer, W. and Christensen, J. P. R., ‘On the existence of pathological submeasures and the construction of exotic topological groups’, Math. Ann. 213 (1975), 203–210.Google Scholar
Johnson, W. B. and Randrianarivony, N. L., ‘ ℓ p (p > 2) does not coarsely embed into a Hilbert space’, Proc. Amer. Math. Soc. 134(4) (2006), 1045–1050.Google Scholar
Kalton, N. J., ‘Coarse and uniform embeddings into reflexive spaces’, Quart. J. Math. 58(3) (2007), 393–414.Google Scholar
Kalton, N. J., ‘The non-linear geometry of Banach spaces’, Rev. Mat. Complut. 21(1) (2008), 7–60.Google Scholar
Kalton, N. J., ‘The uniform structure of Banach spaces’, Math. Ann. 354(4) (2012), 1247–1288.Google Scholar
Kechris, A. S. and Rosendal, C., ‘Turbulence, amalgamation and generic automorphisms of homogeneous structures’, Proc. Lond. Math. Soc. (3) 94(2) (2007), 302–350.Google Scholar
Krivine, J.-L. and Maurey, B., ‘Espaces de Banach stables’, Israel J. Math. 39(4) (1981), 273–295.Google Scholar
Lusky, W., ‘The Gurarij spaces are unique’, Arch. Math. 27 (1976), 627–635.Google Scholar
Megrelishvili, M. G., ‘Every semitopological semigroup compactification of the group H +[0, 1] is trivial’, Semigroup Forum 63(3) (2001), 357–370.Google Scholar
Megrelishvili, M. G., ‘Operator topologies and reflexive representability’, inNuclear Groups and Lie Groups (Madrid, 1999), Research and Exposition in Mathematics, Vol. 24 (Heldermann, Lemgo, 2001), 197–208.Google Scholar
Mendel, M. and Naor, A., ‘Euclidean quotients of finite metric spaces’, Adv. Math. 189 (2004), 451–494.Google Scholar
Mendel, M. and Naor, A., ‘Metric cotype’, Ann. of Math. (2) 168 (2008), 247–298.Google Scholar
Monod, N., Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Mathematics, 1758 (Springer, Berlin Heidelberg, 2001).Google Scholar
Moore, E. H., ‘On properly positive Hermitian matrices’, Bull. Amer. Math. Soc. (N.S.) 23(59) (1916), 66–67.Google Scholar
Naor, A., ‘Uniform nonextendability from nets’, C. R. Acad. Sci.- Series I - Mathématique 353(11) (2015), 991–994.Google Scholar
Naor, A. and Peres, Y., ‘ L p compression, traveling salesmen, and stable walks’, Duke Math. J. 157(1) (2011), 53–108.Google Scholar
Odell, E. and Schlumprecht, T., ‘The distortion problem’, Acta Math. 173 (1994), 259–281.Google Scholar
Pestov, V., ‘A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space’, Topol. Appl. 155(14) (2008), 1561–1575.Google Scholar
Pisier, G., ‘Martingales with values in uniformly convex spaces’, Israel J. Math. 20 (1975), 326–350.Google Scholar
Randrianarivony, N. L., ‘Characterization of quasi-Banach spaces which coarsely embed into a Hilbert space’, Proc. Amer. Math. Soc. 134(5) (2006), 1315–1317.Google Scholar
Raynaud, Y., ‘Espaces de Banach superstables, distances stables et homéomorphismes uniformes, (French. English summary) [Superstable Banach spaces, stable distances and uniform homeomorphisms]’, Israel J. Math. 44(1) (1983), 33–52.Google Scholar
Roe, J., Lectures on Coarse Geometry, University Lecture Series, 31 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Rolewicz, S., Metric Linear Spaces (Reidel, Dordrecht, 1985).Google Scholar
Rosendal, C., ‘A topological version of the Bergman property’, Forum Math. 21(2) (2009), 299–332.Google Scholar
Rosendal, C., ‘Finitely approximable groups and actions part I: the Ribes–Zalesskiĭ property’, J. Symbolic Logic 76(4) (2011), 1297–1306.Google Scholar
Rosendal, C., ‘Coarse geometry of topological groups’, book manuscript 2017.Google Scholar
Ryll-Nardzewski, C., ‘Generalized random ergodic theorems and weakly almost periodic functions’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 271–275.Google Scholar
Schneider, F. M. and Thom, A., ‘On Følner sets in topological groups’, Preprint, 2016, arXiv:1608.08185.Google Scholar
Shtern, A. I., ‘Compact semitopological semigroups and reflexive representability of topological groups’, Russ. J. Math. Phys. 2(1) (1994), 131–132.Google Scholar
Simonnet, M., Measures and Probabilities (Springer, New york, 1996).Google Scholar
Solecki, S., ‘Extending partial isometries’, Israel J. Math. 150 (2005), 315–332.Google Scholar