Distinguishing properties of Arens irregularity
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- by Zhiguo Hu and Matthias Neufang
- Proc. Amer. Math. Soc. 137 (2009), 1753-1761
- DOI: https://doi.org/10.1090/S0002-9939-08-09678-0
- Published electronically: November 17, 2008
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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
- 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
- C. Auger, Jordan Arens irregularity, Master’s thesis, Carleton University, 2007.
- A. Bouziad ; M. Filali, On the size of quotients of function spaces on a topological group, preprint (2008).
- 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
- C. K. Fong ; M. Neufang, On the quotient space $UC(G)/WAP(G)$ and extreme non-Arens regularity of $L_1(H)$, preprint (2006).
- Brian Forrest, Arens regularity and discrete groups, Pacific J. Math. 151 (1991), no. 2, 217–227. MR 1132386, DOI 10.2140/pjm.1991.151.217
- C. C. Graham, Arens regularity of $H^1$, preprint.
- Edmond 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 336241, DOI 10.1090/S0002-9947-1974-0336241-0
- Edmond E. Granirer, On group representations whose $C^{\ast }$-algebra is an ideal in its von Neumann algebra, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, v, 37–52 (English, with French summary). MR 558587
- Edmond 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), no. 3, 402–419. MR 1407625
- Zhiguo Hu, Extreme non-Arens regularity of quotients of the Fourier algebra $A(G)$, Colloq. Math. 72 (1997), no. 2, 237–249. MR 1426699, DOI 10.4064/cm-72-2-237-249
- Zhiguo Hu, Inductive extreme non-Arens regularity of the Fourier algebra $A(G)$, Studia Math. 151 (2002), no. 3, 247–264. MR 1917836, DOI 10.4064/sm151-3-4
- Zhiguo Hu, Maximally decomposable von Neumann algebras on locally compact groups and duality, Houston J. Math. 31 (2005), no. 3, 857–881. MR 2148807
- Zhiguo Hu, Open subgroups and the centre problem for the Fourier algebra, Proc. Amer. Math. Soc. 134 (2006), no. 10, 3085–3095. MR 2231636, DOI 10.1090/S0002-9939-06-08334-1
- Zhiguo Hu and Matthias Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math. 58 (2006), no. 4, 768–795. MR 2245273, DOI 10.4153/CJM-2006-031-7
- Z. Hu, M. Neufang, ; Z.-J. Ruan, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, preprint.
- Z. Hu, M. Neufang, ; Z.-J. Ruan, On topological centre problems and SIN-quantum groups, preprint.
- 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 To Ming Lau and Ali Ülger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1191–1212. MR 1322952, DOI 10.1090/S0002-9947-96-01499-7
- Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
- V. Losert, The centre of the bidual of Fourier algebras, preprint.
- 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.
- Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
- John S. Pym, The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc. (3) 15 (1965), 84–104. MR 173152, DOI 10.1112/plms/s3-15.1.84
- A. Ülger, Central elements of $A^{\ast \ast }$ for certain Banach algebras $A$ without bounded approximate identities, Glasg. Math. J. 41 (1999), no. 3, 369–377. MR 1720442, DOI 10.1017/S0017089599000385
- A. Ülger, Characterizations of the Riesz sets, preprint.
Bibliographic 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
- MR Author ID: 718390
- Email: mneufang@math.carleton.ca
- 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
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2470834