Projections and approximate identities for ideals in group algebras

Liu, Teng-Sun; van Rooij, Arnoud; Wang, Ju Kwei

doi:10.1090/S0002-9947-1973-0318781-2

Projections and approximate identities for ideals in group algebras

Authors: Teng-Sun Liu, Arnoud van Rooij and Ju Kwei Wang
Journal: Trans. Amer. Math. Soc. 175 (1973), 469-482
MSC: Primary 43A20
DOI: https://doi.org/10.1090/S0002-9947-1973-0318781-2
MathSciNet review: 0318781
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a locally compact group G with property $({{\text{P}}_1})$ , if there is a continuous projection of ${L^1}(G)$ onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2-sided approximate identity. The converse is proved for ${w^ \ast }$ -closed left ideals.

Let G be further abelian and let I be a closed ideal in ${L^1}(G)$ . The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group.

[1] N. Bourbaki, Éléments de mathématique. XXV. Première partie. Livre VI: Intégration. Chapitre 6: Intégration vectorielle, Actualités Sci. Ind. No. 1281, Hermann, Paris, 1959 (French). MR 0124722
[2] Paul J. Cohen, Factorization in group algebras, Duke Math. J 26 (1959), 199–205. MR 0104982
[3] John E. Gilbert, On a strong form of spectral synthesis, Ark. Mat. 7 (1969), 571–575. MR 0278008, https://doi.org/10.1007/BF02590895
[4] John E. Gilbert, On projections of 𝐿^{∞}(𝐺) onto translation-invariant subspaces, Proc. London Math. Soc. (3) 19 (1969), 69–88. MR 0244705, https://doi.org/10.1112/plms/s3-19.1.69
[5] Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
[6] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
[7] Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
[8] -, Sur certains idéaux dans ${L^1}(G)$ , C. R. Acad. Sci. Paris Sér. A--B 267 (1968), A882-A885. MR 39 #6025.
[9] Haskell P. Rosenthal, Projections onto translation-invariant subspaces of 𝐿^{𝑝}(𝐺), Mem. Amer. Math. Soc. No. 63 (1966), 84. MR 0211198
[10] Haskell P. Rosenthal, On the existence of approximate identities in ideals of group algebras., Ark. Mat. 7 (1967), 185–191 (1967). MR 0239362, https://doi.org/10.1007/BF02591035
[11] Walter Rudin, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (1962), 429–432. MR 0138012, https://doi.org/10.1090/S0002-9939-1962-0138012-4
[12] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
[13] Bert M. Schreiber, On the coset ring and strong Ditkin sets, Pacific J. Math. 32 (1970), 805–812. MR 0259502