Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T06:06:24.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong and Extremely Strong Ditkin sets forthe Banach Algebras Apr(G) = Ap ⋂ Lr(G)

Published online by Cambridge University Press:  20 November 2018

Edmond E. Granirer
Edmond E. Granirer*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 email: granirer@math.ubc.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ${{A}_{p}}\left( G \right)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus ${{A}_{2}}\left( G \right)$ is the Fourier algebra. Strong Ditkin $\left( \text{SD} \right)$ and Extremely Strong Ditkin $\left( \text{ESD} \right)$ sets for the Banach algebras $A_{P}^{r}\left( G \right)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that $\text{SD}$ and $\text{ESD}$ sets for ${{A}_{p}}\left( G \right)$ remain $\text{SD}$ and $\text{ESD}$ sets for $A_{P}^{r}\left( G \right)$, with strict inclusion for $\text{ESD}$ sets. The case for the strict inclusion of $\text{SD}$ sets is left open.

A result on the weak sequential completeness of ${{A}_{2}}\left( F \right)$ for $\text{ESD}$ sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not $\text{ESD}$ sets for ${{A}_{2}}\left( G \right)$, yet they are such for $A_{2}^{r}\left( G \right)$ for discrete groups $G$, for any $1\,\le \,r\,\le \,2$.

A result is given on the equivalence of the sequential and the net definitions of $\text{SD}$ or $\text{ESD}$ sets for $\sigma $-compact groups.

The above results are new even if $G$ is abelian.

Keywords

43A1543A1046J1043A45Fourier algebraFiga–Talamanca–Herz algebralocally compact groupDitkin setsHelson setsSidon setsweak sequential completeness
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 123 - 135
Copyright
Copyright © Canadian Mathematical Society 2011

References

[Bu1] Burnham, J. T., Closed ideals in subalgebras of Banach algebras. I. Proc. Amer. Math. Soc. 32 (1972), 551-555.Google Scholar
[Dix] Dixmier, J., Les C¤ algèbres et leur reprèsentations. Deuxième ed., Cahiers Scientifiques, XXIX, Gauthier-Villars, 1969.Google Scholar
[dCH] Cannière, J. de and Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1985), no. 2, 455-500. doi:10.2307/2374423Google Scholar
[Do] Dorofaeff, B., The Fourier algebra of SL(2, R) Orn, n2, has no multiplier bounded approximate unit. Math. Ann. 297 (1993), no. 4, 707-724. doi:10.1007/BF01459526Google Scholar
[DuR] Dunkl, C. F. and Ramirez, D. E., Topics in harmonic analysis. Appleton-Century Mathematics Series, Appleton-Century-Crofts. New York, 1971.Google Scholar
[DS] Dunford, N. and, Schwartz, J. T., Linear operators. I. General theory. Pure and Applied Mathematics, 7, Interscience Publishers, New York, 1958.Google Scholar
[Ey1] Eymard, P., L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181-236.Google Scholar
[Ey2] Eymard, P., Algèbres Ap et convoluteurs de Lp. Sèminaire Bourbaki 1969/70, Lecture Notes in Math., 180, Springer, Berlin, 1971, pp. 364-381.Google Scholar
[Fo1] Forrest, B., Amenability and bounded approximate identities in ideals of A(G). Illinois J. Math. 34 (1990), no. 1, 1-25.Google Scholar
[Fo2] Forrest, B., Amenability and ideals in A(G). J. Austral. Math. Soc. Ser. A 53 (1992), no. 2, 143-155. doi:10.1017/S1446788700035758Google Scholar
[FTP] A., Figà-Talamanca and Picardello, M., Multiplicateurs de A(G) qui ne sont pas dans B(G). C. R. Acad. Sci. Paris Sèr. A-B 277 (1973), A117-A119.Google Scholar
[Gi] Gilbert, R. J., On a strong form of spectral synthesis. Ark. Mat. 7 (1969), 571-575.Google Scholar
[GMc] Graham, C. C. and McGehee, O. C., Essays in commutative harmonic analysis. Grundlehren der MathematischenWissenschaften, 283, Springer-Verlag, New York-Berlin, 1979.Google Scholar
[Grh] Graham, C. C., Local existence of K-sets, projective tensor products, and Arens regularity for A(E1 +… + En). Proc. Amer. Math. Soc. 132 (2004), no. 7, 1963-1971. doi:10.1090/S0002-9939-04-07159-XGoogle Scholar
[Gr1] Granirer, E. E., The Figa-Talamanca-Herz-Lebesgue Banach algebras Ar p(G) = Ap(G) \ Lr(G). Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 3, 401-416. doi:10.1017/S0305004105009163Google Scholar
[HaKr] Haagerup, U. and, Kraus, J., Approximation properties for group C¤ algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344 (1994), no. 2, 667-699. doi:10.2307/2154501Google Scholar
[HR] Hewitt, E. and, Ross, K. A., Abstract harmonic analysis. Vols. 1 and 2. Springer-Verlag, New York-Berlin, 1963, 1970.Google Scholar
[Ho] Host, B., Le thèorème des idempotents dans B(G). Bull. Soc. Math. France 114 (1986), no. 2, 215-223.Google Scholar
[Hz1] Herz, C., Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91-123.Google Scholar
[Hz2] Herz, C., The theory of p spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154 (1971), 69-82.Google Scholar
[Le] Leinert, M., Faltungsoperatoren auf gewissen diskreten Gruppen. Studia Math. 52 (1974), 149-158.Google Scholar
[Lu2] F., Lust-Piquard, Means on CVp(G)-subspaces of CVp(G) with RNP and Schur property. Ann. Inst. Fourier (Grenoble) 39 (1989), no. 4, 969-1006.Google Scholar
[Mi] Miao, T., Approximation properties and approximate identities of Ap(G). Trans. Amer. Math. Soc. 361 (2009), no. 3, 1581-1595. doi:10.1090/S0002-9947-08-04674-6Google Scholar
[MS] L., De-Michele and Soardi, P. M., Existence of Sidon sets in discrete FC-groups. Proc. Amer. Math. Soc. 56 (1976), no. 2, 457-460. doi:10.1090/S0002-9939-1976-0397316-0Google Scholar
[Pic] Picardello, M. A., Lacunary sets in discrete noncommutative groups. Boll. Un. Mat. Ital. (4) 8 (1973), 494-508.Google Scholar
[RS] Reiter, H. and, Stegeman, J. T., Classical harmonic analysis and locally compact groups. London Mathematical Society Monographs, New Series, 22, The Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
[Ro] Rosenthal, H. P., On the existence of approximate identities in ideals of group algebras. Ark. Mat. 7 (1967), 185-191. doi:10.1007/BF02591035Google Scholar
[Rob] Robinson, D. J. S., Finiteness conditions and generalized soluble groups. I. Ergebnisse der Mathemtik und ihrer Grenzgebiete, 62, Springer-Verlag, New York-Berlin, 1972.Google Scholar
[Ru] Rudin, W., Fourier analysis on groups. Interscience Tracts in Pure and Applied Mathematics, 12, Interscience Publishers, New York-London, 1962. 1960.Google Scholar
[Sa] Saeki, S., On strong Ditkin sets. Ark. Mat. 10 (1972), 1-7. doi:10.1007/BF02384797Google Scholar
[Saa1] Saab, E., Some characterizations of weak Radon-Nicodym sets. Proc. Amer. Math. Soc. 86 (1982), no. 2, 307-311.Google Scholar
[Saa2] Saab, E. and, Saab, P., A dual geometric characterisation of Banach spaces not containing. Pacific J. Math. 105 (1983), no. 2, 415-425.Google Scholar
[Sch] Schreiber, B. M., On the coset ring and strong Ditkin sets. Pacific J. Math. 32 (1970), 805-812.Google Scholar
[Tak] Takesaki, M., Theory of operator algebras. I. Springer Verlag, New York, 1979.Google Scholar
[Tho] Thoma, E., Über unitäre Darstellung abzählalbarer, diskreter Gruppen. Math. Ann. 153 (1964), 111-138. doi:10.1007/BF01361180Google Scholar
[Va1] Varopoulos, N. Th., Tensor algebras and harmonic analysis. Acta Math. 119 (1967), 51-112. doi:10.1007/BF02392079Google Scholar
[Va2] Varopoulos, N. Th., Tensor algebras over discrete spaces. J. Functional Analysis 3 (1969), 321-335. doi:10.1016/0022-1236(69)90046-9Google Scholar