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Finite-dimensional invariant subspace property and amenability for a class of Banach algebras


Authors: Anthony To-Ming Lau and Yong Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 3755-3775
MSC (2010): Primary 46H20, 43A20, 43A10; Secondary 46H25, 16E40
DOI: https://doi.org/10.1090/tran/6442
Published electronically: July 1, 2015
MathSciNet review: 3453356
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Abstract: Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author. We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra.


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  • [1] J. W. Baker and M. Filali, On minimal ideals in some Banach algebras associated with a locally compact group, J. London Math. Soc. (2) 63 (2001), no. 1, 83-98. MR 1802759 (2002m:43003), https://doi.org/10.1112/S0024610700001733
  • [2] E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), no. 8, 865-884. MR 2013149 (2004k:46129), https://doi.org/10.1142/S0129167X03002046
  • [3] Walter R. Bloom and Herbert Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Mathematics, vol. 20, Walter de Gruyter & Co., Berlin, 1995. MR 1312826 (96a:43001)
  • [4] Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753 (2002a:46082)
  • [5] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726 (2002e:46001)
  • [6] H. G. Dales, A. T.-M. Lau, and D. Strauss, Second duals of measure algebras, Dissertationes Math. (Rozprawy Mat.) 481 (2012), 1-121. MR 2920625, https://doi.org/10.4064/dm481-0-1
  • [7] Shawn Desaulniers, Rasoul Nasr-Isfahani, and Mehdi Nemati, Common fixed point properties and amenability of a class of Banach algebras, J. Math. Anal. Appl. 402 (2013), no. 2, 536-544. MR 3029168, https://doi.org/10.1016/j.jmaa.2012.12.057
  • [8] Pierre Eymard, Sur les applications qui laissent stable l'ensemble des fonctions presque-périodiques, Bull. Soc. Math. France 89 (1961), 207-222 (French). MR 0139689 (25 #3120)
  • [9] Ky Fan, Invariant subspaces for a semigroup of linear operators, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math. 27 (1965), 447-451. MR 0178367 (31 #2625)
  • [10] Ky Fan, Invariant cross-sections and invariant linear subspaces, Israel J. Math. 2 (1964), 19-26. MR 0171151 (30 #1382)
  • [11] Ky Fan, Invariant subspaces of certain linear operators, Bull. Amer. Math. Soc. 69 (1963), 773-777. MR 0158268 (28 #1494)
  • [12] M. Filali, Finite-dimensional left ideals in some algebras associated with a locally compact group, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2325-2333. MR 1487366 (99j:22005), https://doi.org/10.1090/S0002-9939-99-04793-0
  • [13] M. Filali, The ideal structure of some Banach algebras, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 567-576. MR 1151333 (93e:22005), https://doi.org/10.1017/S0305004100075642
  • [14] M. Filali, M. Neufang, and M. Sangani Monfared, On ideals in the bidual of the Fourier algebra and related algebras, J. Funct. Anal. 258 (2010), no. 9, 3117-3133. MR 2595737 (2011f:22007), https://doi.org/10.1016/j.jfa.2009.12.011
  • [15] M. Filali and M. Sangani Monfared, Finite-dimensional left ideals in the duals of introverted spaces, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3645-3656. MR 2813394 (2012e:46096), https://doi.org/10.1090/S0002-9939-2011-10784-6
  • [16] E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. MR 0197595 (33 #5760)
  • [17] Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549 (40 #4776)
  • [18] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915 (28 #158)
  • [19] Z. Hu, M. Sangani Monfared, and T. Traynor, On character amenable Banach algebras, Studia Math. 193 (2009), no. 1, 53-78. MR 2506414 (2010d:46060), https://doi.org/10.4064/sm193-1-3
  • [20] I. S. Iohvidov, Unitary operators in a space with an indefinite metric, Harkov. Gos. Univ. Uč. Zap. 29 = Zap. Mat. Otd. Fiz.-Mat. Fak. i Harkov. Mat. Obšč. (4) 21 (1949), 79-86 (Russian). MR 0081452 (18,405d)
  • [21] I. S. Iohvidov and M. G. Kreĭn, Spectral theory of operators in spaces with indefinite metric. I, Amer. Math. Soc. Transl. (2) 13 (1960), 105-175. MR 0113145 (22 #3983)
  • [22] M. G. Krein, On an application of the fixed-point principle in the theory of linear transformations of spaces with an indefinite metric, Amer. Math. Soc. Transl. (2) 1 (1955), 27-35. MR 0073956 (17,514b)
  • [23] Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837-934 (English, with English and French summaries). MR 1832993 (2002f:46108), https://doi.org/10.1016/S0012-9593(00)01055-7
  • [24] Anthony To-Ming Lau, Finite dimensional invariant subspace properties and amenability, J. Nonlinear Convex Anal. 11 (2010), no. 3, 587-595. MR 2778679 (2012b:43002)
  • [25] Anthony To Ming Lau, Fixed point and finite-dimensional invariant subspace properties for semigroups and amenability, Nonlinear and convex analysis in economic theory (Tokyo, 1993) Lecture Notes in Econom. and Math. Systems, vol. 419, Springer, Berlin, 1995, pp. 203-213. MR 1354424 (96k:43004), https://doi.org/10.1007/978-3-642-48719-4_16
  • [26] A. T. Lau, Fourier and Fourier-Stieltjes algebras of a locally compact group and amenability, Topological vector spaces, algebras and related areas (Hamilton, ON, 1994), Pitman Res. Notes Math. Ser., vol. 316, Longman Sci. Tech., Harlow, 1994, pp. 79-92. MR 1319375 (96k:43005)
  • [27] Anthony To Ming Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987), 195-205. MR 891287 (88f:43006)
  • [28] Anthony To Ming Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161-175. MR 736276 (85k:43007)
  • [29] Anthony To Ming Lau, Finite-dimensional invariant subspaces for a semigroup of linear operators, J. Math. Anal. Appl. 97 (1983), no. 2, 374-379. MR 723239 (85i:43002), https://doi.org/10.1016/0022-247X(83)90203-2
  • [30] Anthony To Ming Lau and Viktor Losert, The $ C^*$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), no. 1, 1-30. MR 1207935 (94d:22005), https://doi.org/10.1006/jfan.1993.1024
  • [31] Anthony To-Ming Lau and J. Ludwig, Fourier-Stieltjes algebra of a topological group, Adv. Math. 229 (2012), no. 3, 2000-2023. MR 2871165 (2012m:43004), https://doi.org/10.1016/j.aim.2011.12.022
  • [32] A. T. Lau, A. L. T. Paterson, and J. C. S. Wong, Invariant subspace theorems for amenable groups, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 3, 415-430. MR 1015485 (90g:47007), https://doi.org/10.1017/S0013091500004673
  • [33] Anthony T. M. Lau and James C. S. Wong, Invariant subspaces for algebras of linear operators and amenable locally compact groups, Proc. Amer. Math. Soc. 102 (1988), no. 3, 581-586. MR 928984 (89c:43007), https://doi.org/10.2307/2047227
  • [34] Anthony T. M. Lau and James C. S. Wong, Finite-dimensional invariant subspaces for measurable semigroups of linear operators, J. Math. Anal. Appl. 127 (1987), no. 2, 548-558. MR 915077 (89c:43006), https://doi.org/10.1016/0022-247X(87)90129-6
  • [35] Anthony T.-M. Lau and Yong Zhang, Fixed point properties for semigroups of nonlinear mappings and amenability, J. Funct. Anal. 263 (2012), no. 10, 2949-2977. MR 2973331, https://doi.org/10.1016/j.jfa.2012.07.013
  • [36] Anthony To-Ming Lau and Yong Zhang, Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal. 254 (2008), no. 10, 2534-2554. MR 2406686 (2010g:47105), https://doi.org/10.1016/j.jfa.2008.02.006
  • [37] Mehdi Sangani Monfared, Character amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 697-706. MR 2418712 (2009b:46104), https://doi.org/10.1017/S0305004108001126
  • [38] Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630-641. MR 0270356 (42 #5245)
  • [39] Theodore Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244-261. MR 0193523 (33 #1743)
  • [40] M. A. Naimark, On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged) 24 (1963), 177-189. MR 0161158 (28 #4367)
  • [41] M. A. Naimark, Commutative unitary operators in the space $ \Pi _{\kappa }$, Dokl. Akad. Nauk SSSR 149 (1963), 1261-1263 (Russian); English transl., Soviet Math. Dokl. 4 (1963), 543-545. MR 0152870 (27 #2842)
  • [42] Jean-Paul Pier, Amenable Banach algebras, Pitman Research Notes in Mathematics Series, vol. 172, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1988. MR 942218 (89g:46093)
  • [43] L. Pontrjagin, Hermitian operators in spaces with indefinite metric, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 8 (1944), 243-280 (Russian, with English summary). MR 0012200 (6,273e)
  • [44] Zhong-Jin Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466-499. MR 1402773 (98e:46077), https://doi.org/10.1006/jfan.1996.0093
  • [45] Zhong-Jin Ruan, The operator amenability of $ A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449-1474. MR 1363075 (96m:43001), https://doi.org/10.2307/2375026
  • [46] Shôichirô Sakai, $ C^*$-algebras and $ W^*$-algebras, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. MR 0442701 (56 #1082)
  • [47] Mahatheva Skantharajah, Amenable hypergroups, Illinois J. Math. 36 (1992), no. 1, 15-46. MR 1133768 (92k:43002)
  • [48] Dan Voiculescu, Amenability and Katz algebras, Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977) Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 451-457. MR 560656 (83c:46065)
  • [49] Benjamin Willson, Invariant nets for amenable groups and hypergroups, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)-University of Alberta (Canada). MR 3122097
  • [50] Benjamin Willson, Configurations and invariant nets for amenable hypergroups and related algebras, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5087-5112. MR 3240918, https://doi.org/10.1090/S0002-9947-2014-05731-0
  • [51] James C. S. Wong, Topological invariant means on locally compact groups and fixed points., Proc. Amer. Math. Soc. 27 (1971), 572-578. MR 0272954 (42 #7835)
  • [52] Yong Zhang, Solved and unsolved problems in generalized notions of amenability for Banach algebras, Banach algebras 2009, Banach Center Publ., vol. 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 441-454. MR 2777469 (2012f:46083), https://doi.org/10.4064/bc91-0-26

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Additional Information

Anthony To-Ming Lau
Affiliation: Department of Mathematical and Statistical sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 Canada
Email: tlau@math.ualberta.ca

Yong Zhang
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2 Canada
Email: zhangy@cc.umanitoba.ca

DOI: https://doi.org/10.1090/tran/6442
Keywords: Fixed point property, invariant mean, finite invariant subspace, F-algebra, quantum group, Hopf von Neumann algebra, ideal, module morphism
Received by editor(s): May 31, 2013
Received by editor(s) in revised form: March 5, 2014
Published electronically: July 1, 2015
Additional Notes: The first author was supported by NSERC Grant MS100
The second author was supported by NSERC Grant 238949
Article copyright: © Copyright 2015 American Mathematical Society

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