Topologies on the ideal space of a Banach algebra and spectral synthesis. (English) Zbl 0883.46032

Beckhoff, Ferdinand

Proc. Am. Math. Soc. 125, No. 10, 2859-2866 (1997).

Summary: Let the space \(\text{Id}(A)\) 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 \(\text{Id}(A)\) 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:
\(\inf \{|\varphi_1-\varphi_2|;\varphi_1,\varphi_2 \in \Delta(A), \varphi_1\neq \varphi_2\}>0\).

MSC:

46J20

Ideals, maximal ideals, boundaries

Keywords:

closed two-sided ideals; normality; weak topology; commutative Banach algebra; spectral synthesis; Gelfand space

References:

[1]	R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987), no. 3, 524 – 542. · Zbl 0613.46048
[2]	F. Beckhoff, Topologies on the space of ideals of a Banach algebra, Studia Mathematica 115 (2) (1995), 189-205. CMP 95:17 · Zbl 0836.46038
[3]	Ferdinand Beckhoff, Topologies of compact families on the ideal space of a Banach algebra, Studia Math. 118 (1996), no. 1, 63 – 75. · Zbl 0854.46045
[4]	Arne Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 1 – 32. · Zbl 0133.07501
[5]	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). · Zbl 0828.46063
[6]	Jacques Dixmier, \?*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. · Zbl 0372.46058
[7]	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. · Zbl 0830.43001
[8]	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). · Zbl 0567.46025
[9]	Simeon Ivanov , 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. · Zbl 0743.00037
[10]	Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. · Zbl 0105.09504
[11]	Donald R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240 – 272. · Zbl 0121.10204
[12]	Douglas W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 39 – 52. · Zbl 0818.46054
[13]	Charles Stegall, A proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 78 (1980), no. 1, 154 – 156. · Zbl 0435.46020
[14]	M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481. · Zbl 0017.13502