Interpolation sets and the size of quotients of function spaces on a locally compact group

Filali, Mahmoud; Galindo, Jorge

doi:10.1090/tran6662

Interpolation sets and the size of quotients of function spaces on a locally compact group
HTML articles powered by AMS MathViewer

by Mahmoud Filali and Jorge Galindo PDF

Trans. Amer. Math. Soc. 369 (2017), 575-603 Request permission

Abstract:

We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation set we introduced and studied recently and unifies the approaches followed by many authors to obtain particular cases.

We find in this way that there is a linear isometric copy of $\ell _\infty (\kappa )$ in each of the following quotient spaces:

[–] $\mathscr {WAP}_0(G)/C_0(G)$ whenever $G$ contains a subset $X$ that is an $E$-set (see the definition in the paper) and $\kappa =\kappa (X)$ is the minimal number of compact sets required to cover $X$. In particular, $\kappa =\kappa (G)$ when $G$ is an $SIN$-group.

[–] $\mathscr {WAP}(G)/\mathscr {B}(G)$, when $G$ is any locally compact group and $\kappa =\kappa (Z(G))$ and $Z(G)$ is the centre of $G$, or when $G$ is either an $IN$-group or a nilpotent group and $\kappa =\kappa (G)$.

[–] $\mathscr {WAP}_0(G)/\mathscr {B}_0(G)$, when $G$ and $\kappa$ are as in the foregoing item.

[–] $\mathscr {CB}(G)/\mathscr {LUC}(G)$, when $G$ is any locally compact group that is neither compact nor discrete and $\kappa =\kappa (G)$.

References

Ofelia T. Alas, Topological groups and uniform continuity, Portugal. Math. 30 (1971), 137–143. MR 302811
Alexander Arhangel′skii and Mikhail Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2433295, DOI 10.2991/978-94-91216-35-0
J. W. Baker and R. J. Butcher, The Stone-Čech compactification of a topological semigroup, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 1, 103–107. MR 409711, DOI 10.1017/S0305004100052725
J. W. Baker and M. Filali, On the analogue of Veech’s theorem in the WAP-compactification of a locally compact group, Semigroup Forum 65 (2002), no. 1, 107–112. MR 1903557, DOI 10.1007/s002330010084
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
Ahmed Bouziad and Mahmoud Filali, On the size of quotients of function spaces on a topological group, Studia Math. 202 (2011), no. 3, 243–259. MR 2771653, DOI 10.4064/sm202-3-3
A. Bouziad and M. Filali, The Stone-Čech compactification of a topological group as a semigroup and the SIN property, Houston J. Math. 38 (2012), no. 4, 1329–1341. MR 3019039, DOI 10.1111/j.1540-6261.1983.tb02305.x
Talin Budak, Nilgün Işık, and John Pym, Minimal determinants of topological centres for some algebras associated with locally compact groups, Bull. Lond. Math. Soc. 43 (2011), no. 3, 495–506. MR 2820139, DOI 10.1112/blms/bdq116
R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
Ching Chou, Weakly almost periodic functions and almost convergent functions on a group, Trans. Amer. Math. Soc. 206 (1975), 175–200. MR 394062, DOI 10.1090/S0002-9947-1975-0394062-8
Ching Chou, Uniform closures of Fourier-Stieltjes algebras, Proc. Amer. Math. Soc. 77 (1979), no. 1, 99–102. MR 539638, DOI 10.1090/S0002-9939-1979-0539638-9
Ching Chou, Minimally weakly almost periodic groups, J. Functional Analysis 36 (1980), no. 1, 1–17. MR 568972, DOI 10.1016/0022-1236(80)90103-2
Ching Chou, Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), no. 1, 141–157. MR 670924, DOI 10.1090/S0002-9947-1982-0670924-2
Ching Chou, Weakly almost periodic functions and thin sets in discrete groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 333–346. MR 984855, DOI 10.1090/S0002-9947-1990-0984855-6
Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870. MR 143056
W. W. Comfort and Kenneth A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. MR 207886
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
Stephen William Drury, Sur les ensembles de Sidon, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A162–A163 (French). MR 271647
Charles F. Dunkl and Donald E. Ramirez, Topics in harmonic analysis, The Appleton-Century Mathematics Series, Appleton-Century-Crofts [Meredith Corporation], New York, 1971. MR 0454515
Heneri A. M. Dzinotyiweyi, Nonseparability of quotient spaces of function algebras on topological semigroups, Trans. Amer. Math. Soc. 272 (1982), no. 1, 223–235. MR 656487, DOI 10.1090/S0002-9947-1982-0656487-6
Ryszard Engelking, General topology, Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60], PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author. MR 0500780
Mahmoud Filali, $t$-sets and some algebraic properties in $\beta S$ and in $l_\infty (S)^\ast$, Semigroup Forum 65 (2002), no. 2, 285–300. MR 1911731, DOI 10.1007/s002330010111
M. Filali, On the actions of a locally compact group on some of its semigroup compactifications, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 25–39. MR 2340973, DOI 10.1017/S0305004107000242
M. Filali and J. Galindo, Approximable ${\scr {WAP}}$- and ${\scr {LUC}}$-interpolation sets, Adv. Math. 233 (2013), 87–114. MR 2995666, DOI 10.1016/j.aim.2012.09.018
M. Filali ; J. Galindo, Algebraic structure of semigroup compactifications: Pym’s and Veech’s Theorems and strongly prime points, Preprint, 2015.
M. Filali ; J. Galindo, Extreme non-Arens regularity of the group algebra, Preprint, 2013.
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 Tero Vedenjuoksu, The Stone-Čech compactification of a topological group and the $\beta$-extension property, Houston J. Math. 36 (2010), no. 2, 477–488. MR 2661257
J. Galindo, On group and semigroup compactifications of topological groups. Manuscript in preparation.
T. E. Gantner, Some corollaries to the metrization lemma, Amer. Math. Monthly 76 (1969), 45–47. MR 238257, DOI 10.2307/2316789
Colin C. Graham and Kathryn E. Hare, Interpolation and Sidon sets for compact groups, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2013. MR 3025283, DOI 10.1007/978-1-4614-5392-5
Edmond E. Granirer, Exposed points of convex sets and weak sequential convergence, Memoirs of the American Mathematical Society, No. 123, American Mathematical Society, Providence, R.I., 1972. Applications to invariant means, to existence of invariant measures for a semigroup of Markov operators etc. . MR 0365090
Edmond E. Granirer, The radical of $L^{\infty }(G)^{\ast }$, Proc. Amer. Math. Soc. 41 (1973), 321–324. MR 326302, DOI 10.1090/S0002-9939-1973-0326302-9
Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 284541, DOI 10.1515/crll.1971.246.1
S. L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math. 18 (1966), 121–137. MR 194912
Kathryn E. Hare and L. Thomas Ramsey, The ubiquity of Sidon sets that are not $I_0$, Preprint, 2016. Available at: http://arxiv.org/pdf/1602.04241.pdf
E. Hewitt ; K. A. Ross, Abstract harmonic analysis I, Springer-Verlag, Berlin, (1963).
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 1998. Theory and applications. MR 1642231, DOI 10.1515/9783110809220
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85–91. MR 50264, DOI 10.4064/fm-38-1-85-91
M. Katĕtov, Correction to “On real-valued functions in topological spaces” (Fund. Math. 38 (1951), pp. 85–91), Fund. Math. 40, (1953) 203–205.
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 John Pym, The topological centre of a compactification of a locally compact group, Math. Z. 219 (1995), no. 4, 567–579. MR 1343662, DOI 10.1007/BF02572381
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
K. de Leeuw and I. Glicksberg, Almost periodic functions on semigroups, Acta Math. 105 (1961), 99–140. MR 131785, DOI 10.1007/BF02559536
K. de Leeuw and I. Glicksberg, The decomposition of certain group representations, J. Analyse Math. 15 (1965), 135–192. MR 186755, DOI 10.1007/BF02787692
B. H. Neumann, Groups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), 227–242 (1955). MR 72138
John Pym, A note on $G^{\scr {LUC}}$ and Veech’s theorem, Semigroup Forum 59 (1999), no. 2, 171–174. MR 1847282, DOI 10.1007/s002339900039
Donald E. Ramirez, Weakly almost periodic functions and Fourier-Stieltjes transforms, Proc. Amer. Math. Soc. 19 (1968), 1087–1088. MR 232162, DOI 10.1090/S0002-9939-1968-0232162-5
Haskell P. Rosenthal, On injective Banach spaces and the spaces $L^{\infty }(\mu )$ for finite measure $\mu$, Acta Math. 124 (1970), 205–248. MR 257721, DOI 10.1007/BF02394572
Walter Rudin, Weak almost periodic functions and Fourier-Stieltjes transforms, Duke Math. J. 26 (1959), 215–220. MR 102705
W. A. F. Ruppert, On weakly almost periodic sets, Semigroup Forum 32 (1985), no. 3, 267–281. MR 815876, DOI 10.1007/BF02575544
Wolfgang Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, vol. 1079, Springer-Verlag, Berlin, 1984. MR 762985, DOI 10.1007/BFb0073675
William A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), no. 5, 775–830. MR 467705, DOI 10.1090/S0002-9904-1977-14319-X
William A. Veech, Weakly almost periodic functions on semisimple Lie groups, Monatsh. Math. 88 (1979), no. 1, 55–68. MR 550072, DOI 10.1007/BF01305857
N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59–62. MR 320756, DOI 10.1093/qmath/24.1.59