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

Studia Mathematica (1996)

• Volume: 118, Issue: 1, page 63-75
• ISSN: 0039-3223

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## Abstract

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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 $U\left(K\right):=I\in Id\left(A\right):I\cap K=\varnothing$, 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 $limin{f}_{i}\parallel x+{I}_{i}\parallel >0$. (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 $\overline{I\cap S}=I$. Examples are separable C*-algebras, the convolution algebras ${L}^{p}\left(G\right)$ where 1 ≤ p < ∞ and G is a metrizable compact group, and others; but not all separable Banach algebras share this property.

## How to cite

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Beckhoff, Ferdinand. "Topologies of compact families on the ideal space of a Banach algebra." Studia Mathematica 118.1 (1996): 63-75. <http://eudml.org/doc/216264>.

@article{Beckhoff1996,
abstract = {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 $U(K) := \{I ∈ Id(A): I ∩ K = ∅\}$, K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if $I_i → I$ in (Id(A),τ()) and x ∈ A╲I imply $lim inf_i∥x + I_i∥ > 0$. (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 $\overline\{I ∩ S\} = I$. 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.},
author = {Beckhoff, Ferdinand},
journal = {Studia Mathematica},
keywords = {locally compact; hull kernel topology; second countable; countable compact sets; normal; separable -algebras; convolution algebras; separable Banach algebras},
language = {eng},
number = {1},
pages = {63-75},
title = {Topologies of compact families on the ideal space of a Banach algebra},
url = {http://eudml.org/doc/216264},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Beckhoff, Ferdinand
TI - Topologies of compact families on the ideal space of a Banach algebra
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 1
SP - 63
EP - 75
AB - 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 $U(K) := {I ∈ Id(A): I ∩ K = ∅}$, K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if $I_i → I$ in (Id(A),τ()) and x ∈ A╲I imply $lim inf_i∥x + I_i∥ > 0$. (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 $\overline{I ∩ S} = I$. 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.
LA - eng
KW - locally compact; hull kernel topology; second countable; countable compact sets; normal; separable -algebras; convolution algebras; separable Banach algebras
UR - http://eudml.org/doc/216264
ER -

## References

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1. [1] R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987), 524-542 Zbl0613.46048
2. [2] F. Beckhoff, Topologies on the space of ideals of a Banach algebra, Studia Math. 115 (1995), 189-205. Zbl0836.46038
3. [3] B. Blackadar, Weak expectations and nuclear C*-algebras, Indiana Univ. Math. J. 27 (1978), 1021-1026 Zbl0393.46047
4. [4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer, 1970. Zbl0213.40103
5. [5] J. L. Kelley, General Topology, Springer, 1955. Zbl0066.16604
6. [6] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52.

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