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Averaged projections, angles between groups and strengthening of Banach property (T). (English) Zbl 1475.22014

Summary: Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties and Banach expanders. In this paper, we introduce a new strengthening of Banach property (T), called “robust Banach property (T)”, which is weaker than strong Banach property (T), but is still strong enough to ensure similar applications. Using the method of averaged projections in Banach spaces and introducing a new notion of angles between projections, we establish a criterion for robust Banach property (T) and show several examples of groups in which this criterion is fulfilled. We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.

MSC:

22D12 Other representations of locally compact groups
46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
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[1] Badea, C., Grivaux, S., Müller, V.: The rate of convergence in the method of alternating projections. Algebra i Analiz 23(3), 1-30 (2011). doi:10.1090/S1061-0022-2012-01202-1 · Zbl 1294.47026 · doi:10.1090/S1061-0022-2012-01202-1
[2] Bader, U., Furman, A., Gelander, T., Monod, N.: Property (T) and rigidity for actions on Banach spaces. Acta Math. 198(1), 57-105 (2007). doi:10.1007/s11511-007-0013-0 · Zbl 1162.22005 · doi:10.1007/s11511-007-0013-0
[3] Bader, U., Gelander, T., Monod, N.: A fixed point theorem for \[L^1\] L1 spaces. Invent. Math. 189(1), 143-148 (2012). doi:10.1007/s00222-011-0363-2 · Zbl 1247.46007 · doi:10.1007/s00222-011-0363-2
[4] Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3(4), 459-470 (1977) · Zbl 0383.47035
[5] Druţu, C., Nowak, P.: Kazhdan projections, random walks and ergodic theorems. arXiv:1501.03473 · Zbl 1429.46036
[6] Dymara, J., Januszkiewicz, T.: Cohomology of buildings and their automorphism groups. Invent. Math. 150(3), 579-627 (2002). doi:10.1007/s00222-002-0242-y · Zbl 1140.20308 · doi:10.1007/s00222-002-0242-y
[7] Ershov, M., Jaikin-Zapirain, A.: Property (T) for noncommutative universal lattices. Invent. Math. 179(2), 303-347 (2010). doi:10.1007/s00222-009-0218-2 · Zbl 1205.22003 · doi:10.1007/s00222-009-0218-2
[8] Kassabov, M.: Subspace arrangements and property T. Groups Geom. Dyn. 5(2), 445-477 (2011). doi:10.4171/GGD/134 · Zbl 1244.20041 · doi:10.4171/GGD/134
[9] de Laat, T., de la Salle, M.: Strong property (T) for higher-rank simple Lie groups. Proc. Lond. Math. Soc. (3) 111(4), 936-966 (2015). doi:10.1112/plms/pdv040 · Zbl 1328.22003
[10] Lafforgue, V.: Un renforcement de la propriété (T). Duke Math. J. 143(3), 559-602 (2008). doi:10.1215/00127094-2008-029 · Zbl 1158.46049 · doi:10.1215/00127094-2008-029
[11] Lafforgue, V.: Propriété (T) renforcée banachique et transformation de Fourier rapide. J. Topol. Anal. 1(3), 191-206 (2009). doi:10.1142/S1793525309000163 · Zbl 1186.46022 · doi:10.1142/S1793525309000163
[12] Liao, B.: Strong Banach property (T) for simple algebraic groups of higher rank. J. Topol. Anal. 6(1), 75-105 (2014). doi:10.1142/S1793525314500010 · Zbl 1291.22010 · doi:10.1142/S1793525314500010
[13] Mimura, M.: Strong algebraization of fixed point properties (2014). arXiv:1505.06728 · Zbl 1244.20041
[14] Ostrovskiĭ, M.I.: Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry. Quaest. Math. 17(3), 259-319 (1994) · Zbl 0807.46014 · doi:10.1080/16073606.1994.9631766
[15] Pietsch, A.: History of Banach spaces and linear operators. Birkhäuser Boston Inc, Boston (2007) · Zbl 1121.46002
[16] Pisier, G.: Some applications of the complex interpolation method to Banach lattices. J. Analyse Math. 35, 264-281 (1979). doi:10.1007/BF02791068 · Zbl 0427.46048 · doi:10.1007/BF02791068
[17] Pisier, G., Xu, Q.H.: Random series in the real interpolation spaces between the spaces \[v_p\] vp. In: Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267. Springer, Berlin, pp. 185-209 (1987). doi:10.1007/BFb0078146 · Zbl 1162.22005
[18] Pustylnik, E., Reich, S., Zaslavski, A.J.: Inner inclination of subspaces and infinite products of orthogonal projections. J. Nonlinear Convex Anal. 14(3), 423-436 (2013) · Zbl 1276.41028
[19] Salle, Md.l.: Towards strong Banach property (T) for SL \[(3, {\mathbb{R}}\] R). Israel J. Math. 211(1), 105-145 (2016). doi:10.1007/s11856-015-1262-9 · Zbl 1362.46028
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