Complemented ideals of group algebras

Andrew Kepert

Studia Mathematica (1994)

  • Volume: 111, Issue: 2, page 123-152
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Kepert, Andrew. "Complemented ideals of group algebras." Studia Mathematica 111.2 (1994): 123-152. <http://eudml.org/doc/216124>.

@article{Kepert1994,
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.},
author = {Kepert, Andrew},
journal = {Studia Mathematica},
keywords = {continuous projection; locally compact abelian group; Banach space complement; dual group; coset ring; elementary sets; elementary chain of projections},
language = {eng},
number = {2},
pages = {123-152},
title = {Complemented ideals of group algebras},
url = {http://eudml.org/doc/216124},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Kepert, Andrew
TI - Complemented ideals of group algebras
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 2
SP - 123
EP - 152
AB - 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.
LA - eng
KW - continuous projection; locally compact abelian group; Banach space complement; dual group; coset ring; elementary sets; elementary chain of projections
UR - http://eudml.org/doc/216124
ER -

References

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  1. [1] 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. 
  2. [2] D. E. Alspach, Complemented translation-invariant subspaces, in: Lecture Notes in Math. 1332, Springer, 1988, 112-125. 
  3. [3] D. E. Alspach and A. Matheson, Projections onto translation-invariant subspaces of L 1 ( ℝ ) , Trans. Amer. Math. Soc. (2) 277 (1983), 815-823. Zbl0516.46013
  4. [4] 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. Zbl0581.43003
  5. [5] D. E. Alspach, A. Matheson and J. M. Rosenblatt, Separating sets by Fourier-Stieltjes transforms, ibid. 84 (1989), 297-311. Zbl0683.43003
  6. [6] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
  7. [7] J. E. Gilbert, On projections of L ∞ ( G ) onto translation-invariant subspaces, Proc. London Math. Soc. (3) 19 (1969), 69-88. Zbl0176.11603
  8. [8] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin, 1963. Zbl0115.10603
  9. [9] D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin, 1932. 
  10. [10] A. G. Kepert, The range of group algebra homomorphisms, ANU Mathematics Research Reports 022-91, SMS-089-91 (1991). 
  11. [11] 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. Zbl0269.22003
  12. [12] H. P. Rosenthal, Projections onto translation-invariant subspaces of L p ( G ) , Mem. Amer. Math. Soc. 63 (1966). Zbl0203.43903
  13. [13] H. P. Rosenthal, On the existence of approximate identities in ideals of group algebras, Ark. Mat. 7 (1967), 185-191. Zbl0172.18303
  14. [14] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962. Zbl0107.09603

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