Finite-dimensional left ideals in the duals of introverted spaces
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- by M. Filali and M. Sangani Monfared PDF
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Abstract:
We use representations of a Banach algebra $A$ to completely characterize all finite-dimensional left ideals in the dual of introverted subspaces of $A^*$ and in particular in the double dual $A^{**}$. We give sufficient conditions under which such ideals always exist and are direct sums of one-dimensional left ideals.References
- Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
- W. G. Bade, H. G. Dales, and Z. A. Lykova, Algebraic and strong splittings of extensions of Banach algebras, Mem. Amer. Math. Soc. 137 (1999), no. 656, viii+113. MR 1491607, DOI 10.1090/memo/0656
- J. W. Baker and M. Filali, Minimal ideals in group algebras and their biduals, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 3, 475–488. MR 1388201, DOI 10.1017/S0305004100075022
- 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
- John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
- 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 and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836, vi+191. MR 2155972, DOI 10.1090/memo/0836
- Antoine Derighetti, Mahmoud Filali, and Mehdi Sangani Monfared, On the ideal structure of some Banach algebras related to convolution operators on $L^p(G)$, J. Funct. Anal. 215 (2004), no. 2, 341–365. MR 2151297, DOI 10.1016/j.jfa.2004.02.011
- Jacques Dixmier, Les $C^*$-algèbres et leurs représentations, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Paris, 1996 (French). Reprint of the second (1969) edition. MR 1452364
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
- 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, 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
- —, Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications, Math. Annalen (to appear).
- M. Filali and J. S. Pym, Right cancellation in the $\scr L\scr U\scr C$-compactification of a locally compact group, Bull. London Math. Soc. 35 (2003), no. 1, 128–134. MR 1935082, DOI 10.1112/S0024609302001649
- Mahmoud Filali and Pekka Salmi, One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras, J. Lond. Math. Soc. (2) 75 (2007), no. 1, 35–46. MR 2302728, DOI 10.1112/jlms/jdl004
- Mahmoud Filali and Pekka Salmi, Slowly oscillating functions in semigroup compactifications and convolution algebras, J. Funct. Anal. 250 (2007), no. 1, 144–166. MR 2345910, DOI 10.1016/j.jfa.2007.05.004
- Fereidoun Ghahramani and Anthony To Ming Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal. 132 (1995), no. 1, 170–191. MR 1346222, DOI 10.1006/jfan.1995.1104
- E. E. Granirer, On some spaces of linear functionals on the algebras $A_p(G)$ for locally compact groups, Colloq. Math. 52 (1987), no. 1, 119–132. MR 891504, DOI 10.4064/cm-52-1-119-132
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Zhiguo Hu, On the set of topologically invariant means on the von Neumann algebra $\textrm {VN}(G)$, Illinois J. Math. 39 (1995), no. 3, 463–490. MR 1339838
- 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
- B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537–539. MR 211260, DOI 10.1090/S0002-9904-1967-11735-X
- Eberhard Kaniuth, Anthony T. Lau, and John Pym, On $\phi$-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 1, 85–96. MR 2388235, DOI 10.1017/S0305004107000874
- Anthony To Ming Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39–59. MR 531968, DOI 10.1090/S0002-9947-1979-0531968-4
- Anthony To Ming Lau, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 267 (1981), no. 1, 53–63. MR 621972, DOI 10.1090/S0002-9947-1981-0621972-9
- Anthony To Ming Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 273–283. MR 817669, DOI 10.1017/S0305004100064197
- 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 and Viktor Losert, On the second conjugate algebra of $L_1(G)$ of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464–470. MR 939122, DOI 10.1112/jlms/s2-37.3.464
- 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 Viktor Losert, The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 27–39. MR 2127225, DOI 10.1017/S0305004104008072
- 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
- Matthias Neufang, A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis, Arch. Math. (Basel) 82 (2004), no. 2, 164–171. MR 2047670, DOI 10.1007/s00013-003-0516-7
- Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000. MR 1736065, DOI 10.1007/978-1-4612-1200-3
- 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
- N. J. Young, Periodicity of functionals and representations of normed algebras on reflexive spaces, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 2, 99–120. MR 435849, DOI 10.1017/S0013091500010610
Additional Information
- M. Filali
- Affiliation: Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
- MR Author ID: 292620
- Email: mahmoud.filali@oulu.fi
- M. Sangani Monfared
- Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada
- MR Author ID: 711290
- Email: monfared@uwindsor.ca
- Received by editor(s): March 31, 2010
- Received by editor(s) in revised form: September 2, 2010
- Published electronically: February 24, 2011
- Additional Notes: The second author was partially supported by NSERC
- Communicated by: Mei-Chi Shaw
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 3645-3656
- MSC (2010): Primary 46H10, 46H15, 46H25, 43A20, 43A60
- DOI: https://doi.org/10.1090/S0002-9939-2011-10784-6
- MathSciNet review: 2813394