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1993 | 106 | 3 | 289-302
Tytuł artykułu

Uniqueness of complete norms for quotients of Banach function algebras

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $Φ_M$.
Słowa kluczowe
Czasopismo
Rocznik
Tom
106
Numer
3
Strony
289-302
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-12-30
poprawiono
1993-04-12
Twórcy
autor
  • Department of Mathematics, University of California, Berkeley, California 94720, U.S.A.
autor
  • Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Bibliografia
  • [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.
  • [2] B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach-Jordan algebras, J. Funct. Anal. 47 (1982), 1-6.
  • [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.
  • [4] W. G. Bade and H. G. Dales, The Wedderburn decomposability of some commutative Banach algebras, J. Funct. Anal. 107 (1992), 105-121.
  • [5] B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539.
  • [6] H. Mirkil, A counterexample to discrete spectral synthesis, Compositio Math. 14 (1960), 269-273.
  • [7] T. J. Ransford, A short proof of Johnson's uniqueness-of-norm theorem, Bull. London Math. Soc. 21 (1989), 487-488.
  • [8] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Math. Monographs, Oxford Univ. Press, 1968.
  • [9] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge Univ. Press, 1976.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv106i3p289bwm
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