Topologies on the space of ideals of a Banach algebra

Ferdinand Beckhoff

Studia Mathematica (1995)

  • Volume: 115, Issue: 2, page 189-205
  • ISSN: 0039-3223

Abstract

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Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely τ ∞ , coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra τ ∞ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if τ ∞ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A) with the relative hull kernel topology turn out to be separable Lindelöf spaces if A is separable, which improves results from [14].

How to cite

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Beckhoff, Ferdinand. "Topologies on the space of ideals of a Banach algebra." Studia Mathematica 115.2 (1995): 189-205. <http://eudml.org/doc/216207>.

@article{Beckhoff1995,
abstract = {Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely $τ_∞$, coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra $τ_∞$ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if $τ_∞$ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A) with the relative hull kernel topology turn out to be separable Lindelöf spaces if A is separable, which improves results from [14].},
author = {Beckhoff, Ferdinand},
journal = {Studia Mathematica},
keywords = {space of two-sided and closed ideals of a Banach algebra; minimal closed primal ideals; Polish space; separable Lindelöf spaces},
language = {eng},
number = {2},
pages = {189-205},
title = {Topologies on the space of ideals of a Banach algebra},
url = {http://eudml.org/doc/216207},
volume = {115},
year = {1995},
}

TY - JOUR
AU - Beckhoff, Ferdinand
TI - Topologies on the space of ideals of a Banach algebra
JO - Studia Mathematica
PY - 1995
VL - 115
IS - 2
SP - 189
EP - 205
AB - Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely $τ_∞$, coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra $τ_∞$ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if $τ_∞$ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A) with the relative hull kernel topology turn out to be separable Lindelöf spaces if A is separable, which improves results from [14].
LA - eng
KW - space of two-sided and closed ideals of a Banach algebra; minimal closed primal ideals; Polish space; separable Lindelöf spaces
UR - http://eudml.org/doc/216207
ER -

References

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  6. [6] F. Beckhoff, The adjunction of a unit and the minimal primal ideal space, in: Proc. 2nd Internat. Conf. in Funct. Anal. and Approx. Theory, Acquafredda di Maratea, September 14-19, 1992, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993), 201-209. Zbl0812.46050
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  9. [9] R. S. Doran and V. A. Belfi, Characterizations of C*-algebras, Marcel Dekker, 1986. Zbl0597.46056
  10. [10] R. A. Hirschfeld and W. Żelazko, On spectral norm Banach algebras, Bull. Acad. Polon. Sci. 16 (1968), 195-199. Zbl0159.18403
  11. [11] W. Rudin, Fourier Analysis on Groups, Interscience, 1962. 
  12. [12] W. Rudin, Functional Analysis, McGraw-Hill, 1973. 
  13. [13] S. Sakai, C*-algebras and W*-algebras, Springer, 1971. 
  14. [14] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52. 
  15. [15] A. Wilansky, Between T 1 and T 2 , Amer. Math. Monthly 74 (1967), 261-266. 

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