Topologies on the space of ideals of a Banach algebra. (English) Zbl 0836.46038

Beckhoff, Ferdinand

Stud. Math. 115, No. 2, 189-205 (1995).

Summary: Some topologies on the space \(\text{Id} (A)\) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely \(\tau_\infty\), coincides with the so-called strong topology if \(A\) is a \(C^*\)-algebra. We prove that for a separable Banach algebra \(\tau_\infty\) coincides with a weaker topology when restricted to the space \(\text{Min-Primal} (A)\) of minimal closed primal ideals and that \(\text{Min-Primal} (A)\) is a Polish space if \(\tau_\infty\) is Hausdorff; this generalizes results from R. J. Archbold [J. Lond. Math. Soc., II. Ser. 35, No. 3, 524-542 (1987; Zbl 0613.46048)]and the author [Mich. Math. J. 40, No. 3, 477-492 (1993; Zbl 0814.46042)]. All subspaces of \(\text{Id} (A)\) with the relative hull kernel topology turn out to be separable Lindelöf spaces if \(A\) is separable, which improves results from D. W. B. Sommerset [Math. Proc. Camb. Philos. Soc. 115, No. 1, 39-52 (1994; Zbl 0818.46054)].

MSC:

46H10	Ideals and subalgebras
46J20	Ideals, maximal ideals, boundaries

Keywords:

space of two-sided and closed ideals of a Banach algebra; minimal closed primal ideals; Polish space; separable Lindelöf spaces

Citations:

Zbl 0613.46048; Zbl 0814.46042; Zbl 0818.46054