# Non-regularity for Banach function algebras

Studia Mathematica (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topFeinstein, 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

top- [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] H. G. Dales and A. M. Davie, Quasianalytic Banach function algebras, J. Funct. Anal. 13 (1973), 28-50. Zbl0254.46027
- [3] J. Dixmier, C*-algebras, North-Holland, Amsterdam, 1982.
- [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] P. Gorkin and R. Mortini, Synthesis sets in ${H}^{\infty}+C$, Indiana Univ. Math. J., to appear.
- [6] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, NJ, 1962. Zbl0117.34001
- [7] J. L. Kelley, General Topology, Van Nostrand, Princeton, NJ, 1955.
- [8] M. M. Neumann, Commutative Banach algebras and decomposable operators, Monatsh. Math. 113 (1992), 227-243. Zbl0765.47010
- [9] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. 1, Cambridge Univ. Press, New York, 1994. Zbl0809.46052
- [10] W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42. Zbl0077.31103
- [11] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, New Delhi, 1974. Zbl0278.26001
- [12] S. Sidney, More on high-order non-local uniform algebras, Illinois J. Math. 18 (1974), 177-192. Zbl0296.46050
- [13] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52.
- [14] D. W. B. Somerset, Ideal spaces of Banach algebras, Proc. London Math. Soc. (3) 78 (1999), 369-400. Zbl1027.46058
- [15] G. Stolzenberg, The maximal ideal space of functions locally in an algebra, Proc. Amer. Math. Soc. 14 (1963), 342-345. Zbl0142.10102
- [16] E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, New York, 1971. Zbl0286.46049
- [17] J. Wermer, Banach algebras and analytic functions, Adv. Math. 1 (1961), 51-102. Zbl0112.07203
- [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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.