Best bounds for the approximate units for certain ideals of 𝐿¹(𝐺) and of 𝐴_{𝑝}(𝐺)

Delaporte, Jacques; Derighetti, Antoine

doi:10.1090/S0002-9939-96-03130-9

Best bounds for the approximate units for certain ideals of $L^{1}(G)$ and of $A_{p}(G)$
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by Jacques Delaporte and Antoine Derighetti PDF

Proc. Amer. Math. Soc. 124 (1996), 1159-1169 Request permission

Abstract:

We compute the best bound for the approximate units of the augmentation ideal of the group algebra $L^{1}(G)$ of a locally compact amenable group $G$. More generally such a calculation is performed for the kernel of the canonical map from $L^{1}(G)$ onto $L^{1}(G/H)$, $H$ being a closed amenable subgroup of $G$. Analogous results involving certain ideals of the Fourier algebra of an amenable group are also discussed.

References

Jacques Delaporte and Antoine Derighetti, On ideals of $A_p$ with bounded approximate units and certain conditional expectations, J. London Math. Soc. (2) 47 (1993), no. 3, 497–506. MR 1214911, DOI 10.1112/jlms/s2-47.3.497
—, $p$-Pseudomeasures and closed subgroups, Mh. Math. 119 (1995), 37–47.
Antoine Derighetti, Some remarks on $L^{1}(G)$, Math. Z. 164 (1978), no. 2, 189–194. MR 517155, DOI 10.1007/BF01174825
Antoine Derighetti, Relations entre les convoluteurs d’un groupe localement compact et ceux d’un sous-groupe fermé, Bull. Sci. Math. (2) 106 (1982), no. 1, 69–84 (French, with English summary). MR 654425
Antoine Derighetti, Quelques observations concernant les ensembles de Ditkin d’un groupe localement compact, Monatsh. Math. 101 (1986), no. 2, 95–113 (French, with English summary). MR 843295, DOI 10.1007/BF01298924
Robert S. Doran and Josef Wichmann, Approximate identities and factorization in Banach modules, Lecture Notes in Mathematics, vol. 768, Springer-Verlag, Berlin-New York, 1979. MR 555240
B. Forrest, Amenability and bounded approximate identities in ideals of $A(G)$, Illinois J. Math. 34 (1990), 1–25.
Brian Forrest, Amenability and the structure of the algebras $A_p(G)$, Trans. Amer. Math. Soc. 343 (1994), no. 1, 233–243. MR 1181182, DOI 10.1090/S0002-9947-1994-1181182-5
Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
Carl Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiii, 145–157 (French, with English summary). MR 425511
C. Herz, Asymmetry of norms of convolution operators. II. Nilpotent Lie groups, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 223–230. MR 0487284
N. Lohoué, Estimations $L^{p}$ des coefficients de représentation et opérateurs de convolution, Adv. in Math. 38 (1980), no. 2, 178–221 (French). MR 597197, DOI 10.1016/0001-8708(80)90004-3
Françoise Piquard, Propriétés harmoniques et géométriques des sous-espaces invariants par translation de $L^{\infty }(G)$, Publications Mathématiques d’Orsay, No. 78-04, Université de Paris-Sud, Département de Mathématiques, Orsay, 1978 (French). MR 0511038
Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
Hans Reiter, Sur certains idéaux dans $L^{1}(G)$, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A882–A885 (French). MR 244711
Hans Reiter, $L^{1}$-algebras and Segal algebras, Lecture Notes in Mathematics, Vol. 231, Springer-Verlag, Berlin-New York, 1971. MR 0440280