Topologies of compact families on the ideal space of a Banach algebra

Beckhoff, Ferdinand

Studia Mathematica, Tome 119 (1996), p. 63-75 / Harvested from The Polish Digital Mathematics Library

Résumé

Let be a family of compact sets in a Banach algebra A such that is stable with respect to finite unions and contains all finite sets. Then the sets $U (K) : = I \in I d (A) : I \cap K = \emptyset$ , K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if $I_{i} \to I$ in (Id(A),τ()) and x ∈ A╲I imply $l i m i n f_{i} ∥ x + I_{i} ∥ > 0$ . (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then 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 ⊂ A such that for all closed two-sided ideals I we have $\bar{I \cap S} = I$ . Examples are separable C*-algebras, the convolution algebras $L^{p} (G)$ where 1 ≤ p < ∞ and G is a metrizable compact group, and others; but not all separable Banach algebras share this property.

Publié le : 1996-01-01

Zbl 0854.46045

EUDML-ID : urn:eudml:doc:216264

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     author = {Ferdinand Beckhoff},
     title = {Topologies of compact families on the ideal space of a Banach algebra},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {63-75},
     zbl = {0854.46045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv118i1p63bwm}
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Beckhoff, Ferdinand. Topologies of compact families on the ideal space of a Banach algebra. Studia Mathematica, Tome 119 (1996) pp. 63-75. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv118i1p63bwm/

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