Finite-dimensional invariant subspace property and amenability for a class of Banach algebras
HTML articles powered by AMS MathViewer
- by Anthony To-Ming Lau and Yong Zhang PDF
- Trans. Amer. Math. Soc. 368 (2016), 3755-3775 Request permission
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.References
- 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, DOI 10.1112/S0024610700001733
- 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, DOI 10.1142/S0129167X03002046
- 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, DOI 10.1515/9783110877595
- 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
- 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
- H. G. Dales, A. T.-M. Lau, and D. Strauss, Second duals of measure algebras, Dissertationes Math. 481 (2012), 1–121. MR 2920625, DOI 10.4064/dm481-0-1
- 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, DOI 10.1016/j.jmaa.2012.12.057
- Pierre Eymard, Sur les applications qui laissent stable l’ensemble des fonctions presque-périodiques, Bull. Soc. Math. France 89 (1961), 207–222 (French). MR 139689, DOI 10.24033/bsmf.1565
- 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, DOI 10.1016/S1385-7258(65)50047-0
- Ky Fan, Invariant cross-sections and invariant linear subspaces, Israel J. Math. 2 (1964), 19–26. MR 171151, DOI 10.1007/BF02759730
- Ky Fan, Invariant subspaces of certain linear operators, Bull. Amer. Math. Soc. 69 (1963), 773–777. MR 158268, DOI 10.1090/S0002-9904-1963-11028-9
- 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, DOI 10.1090/S0002-9939-99-04793-0
- M. Filali, The ideal structure of some Banach algebras, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 567–576. MR 1151333, DOI 10.1017/S0305004100075642
- 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, DOI 10.1016/j.jfa.2009.12.011
- 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, DOI 10.1090/S0002-9939-2011-10784-6
- E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177–197. MR 197595, DOI 10.7146/math.scand.a-10772
- 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
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- Z. Hu, M. Sangani Monfared, and T. Traynor, On character amenable Banach algebras, Studia Math. 193 (2009), no. 1, 53–78. MR 2506414, DOI 10.4064/sm193-1-3
- I. S. Iohvidov, Unitary operators in a space with an indefinite metric, Har′kov. Gos. Univ. Uč. Zap. 29 = Zap. Mat. Otd. Fiz.-Mat. Fak. i Har′kov. Mat. Obšč. (4) 21 (1949), 79–86 (Russian). MR 0081452
- 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, DOI 10.1090/trans2/013/06
- 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, DOI 10.1090/trans2/001/02
- 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, DOI 10.1016/S0012-9593(00)01055-7
- Anthony To-Ming Lau, Finite dimensional invariant subspace properties and amenability, J. Nonlinear Convex Anal. 11 (2010), no. 3, 587–595. MR 2778679
- 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, DOI 10.1007/978-3-642-48719-4_{1}6
- 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
- Anthony To Ming Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987), 195–205. MR 891287, DOI 10.4064/cm-51-1-195-205
- 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, DOI 10.4064/fm-118-3-161-175
- 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, DOI 10.1016/0022-247X(83)90203-2
- 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, DOI 10.1006/jfan.1993.1024
- Anthony To-Ming Lau and J. Ludwig, Fourier-Stieltjes algebra of a topological group, Adv. Math. 229 (2012), no. 3, 2000–2023. MR 2871165, DOI 10.1016/j.aim.2011.12.022
- 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, DOI 10.1017/S0013091500004673
- 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, DOI 10.1090/S0002-9939-1988-0928984-8
- 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, DOI 10.1016/0022-247X(87)90129-6
- 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, DOI 10.1016/j.jfa.2012.07.013
- 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, DOI 10.1016/j.jfa.2008.02.006
- Mehdi Sangani Monfared, Character amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 697–706. MR 2418712, DOI 10.1017/S0305004108001126
- Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
- Theodore Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244–261. MR 193523, DOI 10.1090/S0002-9947-1965-0193523-8
- M. A. Naĭmark, On commuting unitary operators in spaces with indefinite metric, Acta Sci. Math. (Szeged) 24 (1963), 177–189. MR 161158
- M. A. Naĭmark, Unitary permutation operators in the space $\Pi _{\kappa }$, Dokl. Akad. Nauk SSSR 149 (1963), 1261–1263 (Russian). MR 0152870
- 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
- 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
- Zhong-Jin Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466–499. MR 1402773, DOI 10.1006/jfan.1996.0093
- Zhong-Jin Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449–1474. MR 1363075, DOI 10.2307/2375026
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
- Mahatheva Skantharajah, Amenable hypergroups, Illinois J. Math. 36 (1992), no. 1, 15–46. MR 1133768
- 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
- Benjamin Willson, Invariant nets for amenable groups and hypergroups, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–University of Alberta (Canada). MR 3122097
- Benjamin Willson, Configurations and invariant nets for amenable hypergroups and related algebras, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5087–5112. MR 3240918, DOI 10.1090/S0002-9947-2014-05731-0
- James C. S. Wong, Topological invariant means on locally compact groups and fixed points, Proc. Amer. Math. Soc. 27 (1971), 572–578. MR 272954, DOI 10.1090/S0002-9939-1971-0272954-X
- 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, DOI 10.4064/bc91-0-26
Additional Information
- Anthony To-Ming Lau
- Affiliation: Department of Mathematical and Statistical sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 Canada
- MR Author ID: 110640
- Email: tlau@math.ualberta.ca
- Yong Zhang
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2 Canada
- ORCID: 0000-0002-0440-6396
- Email: zhangy@cc.umanitoba.ca
- 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 - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3453356