Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Distinguishing properties of Arens irregularity


Authors: Zhiguo Hu and Matthias Neufang
Journal: Proc. Amer. Math. Soc. 137 (2009), 1753-1761
MSC (2000): Primary 43A20, 43A30, 46H05
DOI: https://doi.org/10.1090/S0002-9939-08-09678-0
Published electronically: November 17, 2008
MathSciNet review: 2470834
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we present a number of examples of commutative Banach algebras with various Arens irregularity properties. These examples illustrate in particular that strong Arens irregularity and extreme non-Arens regularity, the two natural concepts of ``maximal'' Arens irregularity for general Banach algebras as introduced by Dales-Lau and Granirer, respectively, are indeed distinct. Thereby, an open question raised by several authors is answered. We also link these two properties to another natural Arens irregularity property.


References [Enhancements On Off] (What's this?)

  • 1. R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848. MR 0045941 (13:659f)
  • 2. C. Auger, Jordan Arens irregularity, Master's thesis, Carleton University, 2007.
  • 3. A. Bouziad - M. Filali, On the size of quotients of function spaces on a topological group, preprint (2008).
  • 4. H. G. Dales, Banach Algebras and Automatic Continuity, London Math. Society Monographs, New Series 24, Oxford University Press, New York, 2000. MR 1816726 (2002e:46001)
  • 5. H. G. Dales - A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836. MR 2155972 (2006k:43002)
  • 6. C. K. Fong - M. Neufang, On the quotient space $ UC(G)/WAP(G)$ and extreme non-Arens regularity of $ L_1(H)$, preprint (2006).
  • 7. B. Forrest, Arens regularity and discrete groups, Pacific J. Math. 151 (1991), 217-227. MR 1132386 (93c:43001)
  • 8. C. C. Graham, Arens regularity of $ H^1$, preprint.
  • 9. E. E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371-382. MR 0336241 (49:1017)
  • 10. E. E. Granirer, On group representations whose $ C^*$-algebra is an ideal in its von Neumann algebra, Ann. Inst. Fourier (Grenoble) 29 (1979), 37-52. MR 558587 (81b:22007)
  • 11. E. E. Granirer, Day points for quotients of the Fourier algebra $ A(G)$, extreme nonergodicity of their duals and extreme non-Arens regularity, Illinois J. Math. 40 (1996), 402-419. MR 1407625 (98c:43005)
  • 12. Z. Hu, Extreme non-Arens regularity of quotients of the Fourier algebra $ A(G)$, Colloq. Math. 72 (1997), 237-249. MR 1426699 (98a:43004)
  • 13. Z. Hu, Inductive extreme non-Arens regularity of the Fourier algebra $ A(G)$, Studia Math. 151 (2002), 247-264. MR 1917836 (2003f:46069)
  • 14. Z. Hu, Maximally decomposable von Neumann algebras on locally compact groups and duality, Houston J. Math. 31 (2005), 857-881. MR 2148807 (2006e:22010)
  • 15. Z. Hu, Open subgroups and the centre problem for the Fourier algebra, Proc. Amer. Math. Soc. 134 (2006), 3085-3095. MR 2231636 (2007e:22004)
  • 16. Z. Hu - M. Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math. 58 (2006), 768-795. MR 2245273 (2008e:46076)
  • 17. Z. Hu, M. Neufang, - Z.-J. Ruan, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, preprint.
  • 18. Z. Hu, M. Neufang, - Z.-J. Ruan, On topological centre problems and SIN-quantum groups, preprint.
  • 19. A. T.-M. Lau - V. Losert, On the second conjugate algebra of $ L\sb 1(G)$ of a locally compact group, J. London Math. Soc. (2) 37 (1988), 464-470. MR 939122 (89e:43007)
  • 20. A. T.-M. Lau - V. Losert, The $ C^*$-algebra generated by operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993), 1-30. MR 1207935 (94d:22005)
  • 21. A. T.-M. Lau - V. Losert, The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups, Math. Proc. Cambridge Philos. Soc. 138 (2005), 27-39. MR 2127225 (2006c:43003)
  • 22. A. T.-M. Lau - A. Ülger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), 1191-1212. MR 1322952 (96h:43003)
  • 23. H. Leptin, Sur l'algèbre de Fourier d'un groupe localement compact, C. R. Acad. Sci. Paris, Sér. A-B 266 (1968), A1180-A1182. MR 0239002 (39:362)
  • 24. V. Losert, The centre of the bidual of Fourier algebras, preprint.
  • 25. V. Losert, On the centre of the bidual of Fourier algebras (the compact case), presentation at the 2004 Istanbul International Conference on Abstract Harmonic Analysis.
  • 26. T. W. Palmer, Banach Algebras and the General Theory of $ *$-Algebras, Volume 1, Cambridge University Press, Cambridge, 1994. MR 1270014 (95c:46002)
  • 27. J. S. Pym. The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc. 15 (1965), 84-104. MR 0173152 (30:3367)
  • 28. A. Ülger, Central elements of $ A^{**}$ for certain Banach algebras $ A$ without bounded approximate identities, Glasg. Math. J. 41 (1999), 369-377. MR 1720442 (2001b:46082)
  • 29. A. Ülger, Characterizations of the Riesz sets, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 43A20, 43A30, 46H05

Retrieve articles in all journals with MSC (2000): 43A20, 43A30, 46H05


Additional Information

Zhiguo Hu
Affiliation: Department of Mathematics and Statistics, University of Windsor, Windsor,Ontario, N9B 3P4, Canada
Email: zhiguohu@uwindsor.ca

Matthias Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario,K1S 5B6, Canada
Email: mneufang@math.carleton.ca

DOI: https://doi.org/10.1090/S0002-9939-08-09678-0
Keywords: Banach algebras, Arens products, topological centres, weakly almost periodic functionals, Fourier algebras.
Received by editor(s): June 16, 2008
Published electronically: November 17, 2008
Additional Notes: Both authors were partially supported by NSERC
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society