Show Summary Details
More options …

# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2017: 0.695
5-year IMPACT FACTOR: 0.750

CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 0.966
Source Normalized Impact per Paper (SNIP) 2017: 0.889

Mathematical Citation Quotient (MCQ) 2016: 0.75

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 5

# Extreme non-Arens regularity of the group algebra

Mahmoud Filali
/ Jorge Galindo
Published Online: 2018-03-01 | DOI: https://doi.org/10.1515/forum-2017-0117

## Abstract

The Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra ${L}^{1}\left(G\right)$ is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that ${L}^{1}\left(G\right)$ is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group G, there is a linear isometry from ${L}^{\mathrm{\infty }}\left(G\right)$ into the quotient space ${L}^{\mathrm{\infty }}\left(G\right)/\mathcal{ℱ}\left(G\right)$, with $\mathcal{ℱ}\left(G\right)$ being any closed subspace of ${L}^{\mathrm{\infty }}\left(G\right)$ made of continuous bounded functions. This, together with the known fact that ${\mathrm{\ell }}^{\mathrm{\infty }}\left(G\right)/\mathcal{𝒲}\mathcal{𝒜}\mathcal{𝒫}\left(G\right)$ always contains a linearly isometric copy of ${\mathrm{\ell }}^{\mathrm{\infty }}\left(G\right)$, proves that ${L}^{1}\left(G\right)$ is extremely non-Arens regular for every infinite locally compact group.

MSC 2010: 22D15; 43A46; 43A15; 43A60; 54H11

## References

• [1]

R. Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19.

• [2]

R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.

• [3]

A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Stud. Math. 1, Atlantis Press, Paris, 2008. Google Scholar

• [4]

J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, Can. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1989. Google Scholar

• [5]

F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb. (3) 80, Springer, Berlin, 1973. Google Scholar

• [6]

A. Bouziad and M. Filali, On the size of quotients of function spaces on a topological group, Studia Math. 202 (2011), no. 3, 243–259.

• [7]

T. Budak, N. Işık and J. Pym, Minimal determinants of topological centres for some algebras associated with locally compact groups, Bull. Lond. Math. Soc. 43 (2011), no. 3, 495–506.

• [8]

C. Chou, Weakly almost periodic functions and Fourier–Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), no. 1, 141–157.

• [9]

P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870.

• [10]

H. G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. New Ser. 24, Oxford University Press, New York, 2000. Google Scholar

• [11]

H. G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836. Google Scholar

• [12]

H. F. Davis, A note on Haar measure, Proc. Amer. Math. Soc. 6 (1955), 318–321.

• [13]

M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. Google Scholar

• [14]

P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236. Google Scholar

• [15]

M. Filali and J. Galindo, Approximable $\mathcal{𝒲}\mathcal{𝒜}\mathcal{𝒫}$- and $\mathcal{ℒ}\mathcal{𝒰}\mathcal{𝒞}$-interpolation sets, Adv. Math. 233 (2013), 87–114. Google Scholar

• [16]

M. Filali and J. Galindo, Interpolation sets and the size of quotients of function spaces on a locally compact group, Trans. Amer. Math. Soc. 369 (2017), no. 1, 575–603. Google Scholar

• [17]

M. Filali and P. Salmi, Slowly oscillating functions in semigroup compactifications and convolution algebras, J. Funct. Anal. 250 (2007), no. 1, 144–166.

• [18]

M. Filali and A. I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras, General Topological Algebras (Tartu 1999), Math. Stud. (Tartu) 1, Estonian Mathematical Society, Tartu (2001), 95–124. Google Scholar

• [19]

M. Filali and T. Vedenjuoksu, Extreme non-Arens regularity of semigroup algebras, Topology Proc. 33 (2009), 185–196. Google Scholar

• [20]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1984. Google Scholar

• [21]

G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math., CRC Press, Boca Raton, 1995. Google Scholar

• [22]

C. K. Fong and M. Neufang, On the quotient space $UC\left(G\right)/WAP\left(G\right)$ and extreme non Arens regularity of ${L}^{1}\left(G\right)$, preprint (2006).

• [23]

B. Forrest, Arens regularity and discrete groups, Pacific J. Math. 151 (1991), no. 2, 217–227.

• [24]

D. H. Fremlin, Measure Theory. Vol. 3: Measure Algebras, Torres Fremlin, Colchester, 2004. Google Scholar

• [25]

E. E. Granirer, Day points for quotients of the Fourier algebra $A\left(G\right)$, extreme nonergodicity of their duals and extreme non-Arens regularity, Illinois J. Math. 40 (1996), no. 3, 402–419. Google Scholar

• [26]

S. Grekas, Isomorphic measures on compact groups, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 2, 349–360.

• [27]

S. Grekas and S. Mercourakis, On the measure-theoretic structure of compact groups, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2779–2796.

• [28]

M. Grosser and V. Losert, The norm-strict bidual of a Banach algebra and the dual of ${C}_{u}\left(G\right)$, Manuscripta Math. 45 (1984), no. 2, 127–146. Google Scholar

• [29]

S. L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math. 18 (1966), 121–137.

• [30]

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963. Google Scholar

• [31]

K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, De Gruyter Stud. Math. 25, Walter de Gruyter, Berlin, 1998. Google Scholar

• [32]

Z. Hu, On the set of topologically invariant means on the von Neumann algebra $\mathrm{VN}\left(G\right)$, Illinois J. Math. 39 (1995), no. 3, 463–490. Google Scholar

• [33]

Z. Hu, Extreme non-Arens regularity of quotients of the Fourier algebra $A\left(G\right)$, Colloq. Math. 72 (1997), no. 2, 237–249. Google Scholar

• [34]

Z. Hu and M. Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math. 58 (2006), no. 4, 768–795.

• [35]

Z. Hu and M. Neufang, Distinguishing properties of Arens irregularity, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1753–1761. Google Scholar

• [36]

N. I̧sik, J. Pym and A. Ülger, The second dual of the group algebra of a compact group, J. Lond. Math. Soc. (2) 35 (1987), no. 1, 135–148. Google Scholar

• [37]

A. T. M. Lau and V. Losert, On the second conjugate algebra of ${L}_{1}\left(G\right)$ of a locally compact group, J. Lond. Math. Soc. (2) 37 (1988), no. 3, 464–470. Google Scholar

• [38]

A. T. M. Lau and J. C. S. Wong, Weakly almost periodic elements in ${L}_{\mathrm{\infty }}\left(G\right)$ of a locally compact group, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1031–1036. Google Scholar

• [39]

V. Losert, Talk at Abstract Harmonic Analysis Conference, Istanbul, 2006.

• [40]

V. Losert, The centre of the bidual of Fourier algebras (discrete groups), Bull. Lond. Math. Soc. 48 (2016), no. 6, 968–976.

• [41]

P. S. Mostert, Sections in principal fibre spaces, Duke Math. J. 23 (1956), 57–71.

• [42]

M. 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.

• [43]

T. W. Palmer, Banach Algebras and the General Theory of *-Algebras. Vol. I: Algebras and Banach Algebras, Encyclopedia Math. Appl. 49, Cambridge University Press, Cambridge, 1994. Google Scholar

• [44]

J. S. Pym, The convolution of functionals on spaces of bounded functions, Proc. Lond. Math. Soc. (3) 15 (1965), 84–104. Google Scholar

• [45]

H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., London Math. Soc. Monogr. New Ser. 22, Oxford University Press, New York, 2000. Google Scholar

• [46]

H. P. Rosenthal, On injective Banach spaces and the spaces ${L}^{\mathrm{\infty }}\left(\mu \right)$ for finite measure μ, Acta Math. 124 (1970), 205–248. Google Scholar

• [47]

H. L. Royden, Real Analysis, Macmillan, New York, 1963. Google Scholar

• [48]

S. Sherman, The second adjoint of a $C*$-algebra, Proceedings of The International Congress of Mathematicians. Vol. 1 (Cambridge 1950), American Mathematical Society, Providence (1952), 470–470. Google Scholar

• [49]

Z. Takeda, Conjugate spaces of operator algebras, Proc. Japan Acad. 30 (1954), 90–95.

• [50]

A. Ülger, Continuity of weakly almost periodic functionals on ${L}^{1}\left(G\right)$, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 495–497. Google Scholar

• [51]

H. Yamabe, A generalization of a theorem of Gleason, Ann. of Math. (2) 58 (1953), 351–365.

• [52]

N. J. Young, Separate continuity and multilinear operations, Proc. Lond. Math. Soc. (3) 26 (1973), 289–319. Google Scholar

• [53]

N. J. Young, The irregularity of multiplication in group algebras, Quart J. Math. Oxford Ser. (2) 24 (1973), 59–62.

Revised: 2018-01-19

Published Online: 2018-03-01

Published in Print: 2018-09-01

Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM2016-77143-P

Parts of the article were written when the first named author was visiting Universitat Jaume I in Castellón in December 2011 and May 2012. He would like to express his warm thanks for the kind hospitality and support. Subsequently, he was partially supported by Väisälä Foundation in 2012–2014. This support is gratefully acknowledged. The second named author was supported by Ministerio de Economía y Competitividad (Spain) through project MTM2016-77143-P (AEI/FEDER, UE). This support is also gratefully acknowledged.

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1193–1208, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.