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Topologies on the ideal space
of a Banach algebra and spectral synthesis


Author: Ferdinand Beckhoff
Journal: Proc. Amer. Math. Soc. 125 (1997), 2859-2866
MSC (1991): Primary 46J20
DOI: https://doi.org/10.1090/S0002-9939-97-03831-8
MathSciNet review: 1389504
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Abstract: Let the space $\id (A)$ of closed two-sided ideals of a Banach algebra $A$ carry the weak topology. We consider the following property called normality (of the family of finite subsets of $A)$: if the net $(I_i)_i$ in $\id (A)$ converges weakly to $I$, then for all $a\in A\backslash I$ we have $\liminf _i\|a+I_i\|>0$ (e.g. $C^*$-algebras, $L^1(G)$ with compact $G,\ldots )$. For a commutative Banach algebra normality is implied by spectral synthesis of all closed subsets of the Gelfand space $\Delta (A)$, the converse does not always hold, but it does under the following additional geometrical assumption: $\rinf \{\|\varphi _1-\varphi _2\|;\varphi _1,\varphi _2 \in \Delta(A), \varphi _1\neq \varphi_2\}>0$.


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  • 1. R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987) 524-542. MR 89h:46076
  • 2. F. Beckhoff, Topologies on the space of ideals of a Banach algebra, Studia Mathematica 115 (2) (1995), 189-205. CMP 95:17
  • 3. F. Beckhoff, Topologies of compact families on the ideal space of a Banach algebra, Studia Mathematica 118 (1) (1996), 63-75. MR 96m:46087
  • 4. A. Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier, Grenoble 14 (1964), 1-32. MR 32:321
  • 5. T. Ceausu, D. Gaspar, Generalized Lipschitz spaces as Banach algebras with spectral synthesis, Analele Universitatii din Timisoara XXX (1992), 173-182, Seria Stiinte Matematice. MR 96b:46070
  • 6. J. Dixmier, $C^*$-algebras, North-Holland-Publishing Company 1977. MR 56:16388
  • 7. E. Hewitt, K. A. Ross, Abstract Harmonic Analysis II, Springer (1970). MR 41:7378
  • 8. L. G. Khanin, Spectral synthesis of ideals in algebras of functions having generalized derivatives, English transl. in Russian Math. Surveys 39 (1984), 167-168. MR 85d:46072
  • 9. L. G. Khanin, The Structure of Closed Ideals in Some Algebras of Smooth Functions, Amer. Math. Soc. Transl. 49 (1991), 97-113. MR 92f:00031
  • 10. W. Rudin, Fourier Analysis on Groups, Interscience Publishers (1962). MR 27:2808
  • 11. D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272. MR 28:4385
  • 12. D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Camb. Philos. Soc. 115 (1994), 39-52. MR 94k:46090
  • 13. C. Stegall, A proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 78, (1980), 154-156. MR 81e:46012
  • 14. M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481.

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Additional Information

Ferdinand Beckhoff
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Email: beckhof@math.uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9939-97-03831-8
Received by editor(s): October 3, 1995
Received by editor(s) in revised form: March 19, 1996
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1997 American Mathematical Society

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