Topologies on the ideal space of a Banach algebra and spectral synthesis
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- by Ferdinand Beckhoff PDF
- Proc. Amer. Math. Soc. 125 (1997), 2859-2866 Request permission
Abstract:
Let the space $\operatorname {Id}(A)$ 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 $\operatorname {Id}(A)$ 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: $\inf \{\|\varphi _1-\varphi _2\|;\varphi _1,\varphi _2 \in \Delta (A), \varphi _1\neq \varphi _2\}>0$.References
- R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987), no. 3, 524–542. MR 918643, DOI 10.1112/jlms/s2-36.3.524
- F. Beckhoff, Topologies on the space of ideals of a Banach algebra, Studia Mathematica 115 (2) (1995), 189–205.
- Ferdinand Beckhoff, Topologies of compact families on the ideal space of a Banach algebra, Studia Math. 118 (1996), no. 1, 63–75. MR 1373625, DOI 10.4064/sm-118-1-63-75
- Arne Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 1–32. MR 182839
- T. Ceauşu and D. Gaşpar, Generalized Lipschitz spaces as Banach algebras with spectral synthesis, An. Univ. Timişoara Ser. Ştiinţ. Mat. 30 (1992), no. 2-3, 173–182 (1993). MR 1330746
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- L. G. Khanin, Spectral synthesis of ideals in algebras of functions having generalized derivatives, Uspekhi Mat. Nauk 39 (1984), no. 2(236), 199–200 (Russian). MR 740037
- Simeon Ivanov (ed.), American Mathematical Society Translations. Series 2. Vol. 149, American Mathematical Society Translations, Series 2, vol. 149, American Mathematical Society, Providence, RI, 1991. Thirteen papers in algebra, functional analysis, topology, and probability, translated from the Russian. MR 1137713, DOI 10.1090/trans2/149
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- Donald R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240–272. MR 161177, DOI 10.1090/S0002-9947-1964-0161177-1
- Douglas W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 39–52. MR 1253281, DOI 10.1017/S0305004100071905
- Charles Stegall, A proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 78 (1980), no. 1, 154–156. MR 548105, DOI 10.1090/S0002-9939-1980-0548105-6
- M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.
Additional Information
- Ferdinand Beckhoff
- Affiliation: Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
- Email: beckhof@math.uni-muenster.de
- Received by editor(s): October 3, 1995
- Received by editor(s) in revised form: March 19, 1996
- Communicated by: Theodore Gamelin
- © Copyright 1997 American Mathematical Society
- 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