Topologies of compact families on the ideal space of a Banach algebra. (English) Zbl 0854.46045

Beckhoff, Ferdinand

Stud. Math. 118, No. 1, 63-75 (1996).

Summary: Let \({\mathcal K}\) be a family of compact sets in a Banach algebra \(A\) such that \({\mathcal K}\) is stable with respect to finite unions and contains all finite sets. Then the sets \(U(K):= \{I\in \text{Id} (A): I\cap K= \emptyset\}\), \(K\in {\mathcal K}\), define a topology \(\tau ({\mathcal K})\) on the space \(\text{Id} (A)\) of closed two-sided ideals of \(A\). \({\mathcal K}\) is called normal if \(I_i\to I\) in \((\text{Id} (A), \tau ({\mathcal K}))\) and \(x\in A\setminus I\) imply \(\liminf_i |x+ I_i|>0\).
(1) If the family of finite subsets of \(A\) is normal then \(\text{Id} (A)\) is locally compact in the hull kernel topology and if moreover \(A\) is separable then \(\text{Id} (A)\) is second countable.
(2) If the family of countable compact sets is normal and \(A\) is separable then there is a countable subset \(S\subset A\) such that for all closed two-sided ideals \(I\) we have \(\overline {I\cap S} =I\).
Examples are separable \(C^*\)-algebras, the convolution algebras \(L^p (G)\) where \(1\leq p< \infty\) and \(G\) is a metrizable compact group, and others; but not all separable Banach algebras share this property.

MSC:

46H10

Ideals and subalgebras

Keywords:

locally compact; hull kernel topology; second countable; countable compact sets; normal; separable \(C^*\)-algebras; convolution algebras; separable Banach algebras