Non-regularity for Banach function algebras

J. Feinstein; D. Somerset

Studia Mathematica (2000)

  • Volume: 141, Issue: 1, page 53-68
  • ISSN: 0039-3223

Abstract

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Let A be a unital Banach function algebra with character space Φ A . For x ∈ Φ A , let M x and J x be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of J x is connected, and that if x does not belong to the Shilov boundary of A then the set y ∈ Φ A : M x ⊇ J y has an infinite connected subset. Various related results are given.

How to cite

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Feinstein, J., and Somerset, D.. "Non-regularity for Banach function algebras." Studia Mathematica 141.1 (2000): 53-68. <http://eudml.org/doc/216773>.

@article{Feinstein2000,
abstract = {Let A be a unital Banach function algebra with character space $Φ_\{A\}$. For $x ∈ Φ_\{A\}$, let $M_\{x\}$ and $J_\{x\}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_\{x\}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $\{y ∈ Φ_\{A\}: M_\{x\} ⊇ J_\{y\}\}$ has an infinite connected subset. Various related results are given.},
author = {Feinstein, J., Somerset, D.},
journal = {Studia Mathematica},
keywords = {Hausdorff property; nonregular Banach function algebras; hulls; ideals},
language = {eng},
number = {1},
pages = {53-68},
title = {Non-regularity for Banach function algebras},
url = {http://eudml.org/doc/216773},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Feinstein, J.
AU - Somerset, D.
TI - Non-regularity for Banach function algebras
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 1
SP - 53
EP - 68
AB - Let A be a unital Banach function algebra with character space $Φ_{A}$. For $x ∈ Φ_{A}$, let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set ${y ∈ Φ_{A}: M_{x} ⊇ J_{y}}$ has an infinite connected subset. Various related results are given.
LA - eng
KW - Hausdorff property; nonregular Banach function algebras; hulls; ideals
UR - http://eudml.org/doc/216773
ER -

References

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  1. [1] R. J. Archbold and C. J. K. Batty, On factorial states of operator algebras, III, J. Operator Theory 15 (1986), 53-81. Zbl0592.46052
  2. [2] H. G. Dales and A. M. Davie, Quasianalytic Banach function algebras, J. Funct. Anal. 13 (1973), 28-50. Zbl0254.46027
  3. [3] J. Dixmier, C*-algebras, North-Holland, Amsterdam, 1982. 
  4. [4] J. F. Feinstein and D. W. B. Somerset, Strong regularity for uniform algebras, in: Proc. 3rd Function Spaces Conference (Edwardsville, IL, 1998), Contemp. Math. 232, Amer. Math. Soc., 1999, 139-149. Zbl0933.46047
  5. [5] P. Gorkin and R. Mortini, Synthesis sets in H ∞ + C , Indiana Univ. Math. J., to appear. 
  6. [6] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, NJ, 1962. Zbl0117.34001
  7. [7] J. L. Kelley, General Topology, Van Nostrand, Princeton, NJ, 1955. 
  8. [8] M. M. Neumann, Commutative Banach algebras and decomposable operators, Monatsh. Math. 113 (1992), 227-243. Zbl0765.47010
  9. [9] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. 1, Cambridge Univ. Press, New York, 1994. Zbl0809.46052
  10. [10] W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42. Zbl0077.31103
  11. [11] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, New Delhi, 1974. Zbl0278.26001
  12. [12] S. Sidney, More on high-order non-local uniform algebras, Illinois J. Math. 18 (1974), 177-192. Zbl0296.46050
  13. [13] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52. 
  14. [14] D. W. B. Somerset, Ideal spaces of Banach algebras, Proc. London Math. Soc. (3) 78 (1999), 369-400. Zbl1027.46058
  15. [15] G. Stolzenberg, The maximal ideal space of functions locally in an algebra, Proc. Amer. Math. Soc. 14 (1963), 342-345. Zbl0142.10102
  16. [16] E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, New York, 1971. Zbl0286.46049
  17. [17] J. Wermer, Banach algebras and analytic functions, Adv. Math. 1 (1961), 51-102. Zbl0112.07203
  18. [18] D. R. Wilken, Approximate normality and function algebras on the interval and the circle, in: Function Algebras (New Orleans, 1965), Scott-Foresman, Chicago, IL, 1966, 98-111. 

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