# 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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topBeckhoff, 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

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

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