Uniqueness of complete norms for quotients of Banach function algebras

W. Bade; H. Dales

Studia Mathematica (1993)

  • Volume: 106, Issue: 3, page 289-302
  • ISSN: 0039-3223

Abstract

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We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra L 1 ( G ) of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients A ( Γ ) / J ( E ) ¯ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct a variety of mutually non-equivalent norms for quotients of the Mirkil algebra M, which fails Ditkin’s condition at only one point of Φ M .

How to cite

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Bade, W., and Dales, H.. "Uniqueness of complete norms for quotients of Banach function algebras." Studia Mathematica 106.3 (1993): 289-302. <http://eudml.org/doc/216018>.

@article{Bade1993,
abstract = {We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra $L^1(G)$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients $A(Γ)/\overline\{J(E)\}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct a variety of mutually non-equivalent norms for quotients of the Mirkil algebra M, which fails Ditkin’s condition at only one point of $Φ_M$.},
author = {Bade, W., Dales, H.},
journal = {Studia Mathematica},
keywords = {quotient algebra of a unital Banach function algebra; unique complete norm; Ditkin algebra; Fourier transforms of the group algebra; sets of non-synthesis; quotients of the Mirkil algebra; Ditkin's condition},
language = {eng},
number = {3},
pages = {289-302},
title = {Uniqueness of complete norms for quotients of Banach function algebras},
url = {http://eudml.org/doc/216018},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Bade, W.
AU - Dales, H.
TI - Uniqueness of complete norms for quotients of Banach function algebras
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 3
SP - 289
EP - 302
AB - We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra $L^1(G)$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients $A(Γ)/\overline{J(E)}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct a variety of mutually non-equivalent norms for quotients of the Mirkil algebra M, which fails Ditkin’s condition at only one point of $Φ_M$.
LA - eng
KW - quotient algebra of a unital Banach function algebra; unique complete norm; Ditkin algebra; Fourier transforms of the group algebra; sets of non-synthesis; quotients of the Mirkil algebra; Ditkin's condition
UR - http://eudml.org/doc/216018
ER -

References

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  1. [1] A. Atzmon, On the union of sets of synthesis and Ditkin's condition in regular Banach algebras, Bull. Amer. Math. Soc. 2 (1980), 317-320. Zbl0432.43004
  2. [2] B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach-Jordan algebras, J. Funct. Anal. 47 (1982), 1-6. Zbl0488.46043
  3. [3] W. G. Bade, P. C. Curtis, Jr. and K. B. Laursen, Automatic continuity in algebras of differentiable functions, Math. Scand. 70 (1977), 249-270. Zbl0373.46061
  4. [4] W. G. Bade and H. G. Dales, The Wedderburn decomposability of some commutative Banach algebras, J. Funct. Anal. 107 (1992), 105-121. Zbl0765.46036
  5. [5] B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539. Zbl0172.41004
  6. [6] H. Mirkil, A counterexample to discrete spectral synthesis, Compositio Math. 14 (1960), 269-273. Zbl0099.10203
  7. [7] T. J. Ransford, A short proof of Johnson's uniqueness-of-norm theorem, Bull. London Math. Soc. 21 (1989), 487-488. Zbl0705.46028
  8. [8] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Math. Monographs, Oxford Univ. Press, 1968. Zbl0165.15601
  9. [9] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge Univ. Press, 1976. Zbl0313.47029

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