Complemented ideals of group algebras

Andrew G. Kepert

ArticleOriginal scientific text

Title

Complemented ideals of group algebras

Authors ^1,²

Affiliations

Matematisk Institut, Københavns Universitet, Universitetsparken 5, DK-2100 København Ø, Denmark
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078-0613, U.S.A.

Abstract

The existence of a projection onto an ideal I of a commutative group algebra

L^{1} (G)

depends on its hull Z(I) ⊆ Ĝ. Existing methods for constructing a projection onto I rely on a decomposition of Z(I) into simpler hulls, which are then reassembled one at a time, resulting in a chain of projections which can be composed to give a projection onto I. These methods are refined and examples are constructed to show that this approach does not work in general. Some answers are also given to previously asked questions concerning such hulls and some conjectures are presented concerning the classification of these complemented ideals.

D. E. Alspach, A characterization of the complemented translation-invariant subspaces of $L^{1} (ℝ^{2})$ , J. London Math. Soc. (2) 31 (1985), 115-124.
D. E. Alspach, Complemented translation-invariant subspaces, in: Lecture Notes in Math. 1332, Springer, 1988, 112-125.
D. E. Alspach and A. Matheson, Projections onto translation-invariant subspaces of $L^{1} (ℝ)$ , Trans. Amer. Math. Soc. (2) 277 (1983), 815-823.
D. E. Alspach, A. Matheson and J. M. Rosenblatt, Projections onto translation-invariant subspaces of $L^{1} (G)$ , J. Funct. Anal. 59 (1984), 254-292; Erratum, ibid. 69 (1986), 141.
D. E. Alspach, A. Matheson and J. M. Rosenblatt, Separating sets by Fourier-Stieltjes transforms, ibid. 84 (1989), 297-311.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973.
J. E. Gilbert, On projections of $L^{\infty} (G)$ onto translation-invariant subspaces, Proc. London Math. Soc. (3) 19 (1969), 69-88.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin, 1963.
D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin, 1932.
A. G. Kepert, The range of group algebra homomorphisms, ANU Mathematics Research Reports 022-91, SMS-089-91 (1991).
T.-S. Liu, A. van Rooij and J.-K. Wang, Projections and approximate identities for ideals in group algebras, Trans. Amer. Math. Soc. 175 (1973), 469-482.
H. P. Rosenthal, Projections onto translation-invariant subspaces of $L^{p} (G)$ , Mem. Amer. Math. Soc. 63 (1966).
H. P. Rosenthal, On the existence of approximate identities in ideals of group algebras, Ark. Mat. 7 (1967), 185-191.
W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.

Title

Complemented ideals of group algebras

Affiliations

Abstract

Bibliography