Commutative, radical amenable Banach algebras

C. Read

Studia Mathematica (2000)

  • Volume: 140, Issue: 3, page 199-212
  • ISSN: 0039-3223

Abstract

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There has been a considerable search for radical, amenable Banach algebras. Noncommutative examples were finally found by Volker Runde [R]; here we present the first commutative examples. Centrally placed within the construction, the reader may be pleased to notice a reprise of the undergraduate argument that shows that a normed space with totally bounded unit ball is finite-dimensional; we use the same idea (approximate the norm 1 vector x within distance η by a “good” vector y 1 ; then approximate ( x - y 1 ) / η within distance η by a “good” vector y 2 , thus approximating x within distance η 2 by y 1 + η y 2 , and so on) to go from η=9/10 in Lemma 1.5 to arbitrarily small η in Lemma 2.1. This is not an arbitrary decision on the part of the author; it really is forced on him by the nature of the construction, see e.g. (6.1) for a place where η small at the start will not do.

How to cite

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Read, C.. "Commutative, radical amenable Banach algebras." Studia Mathematica 140.3 (2000): 199-212. <http://eudml.org/doc/216764>.

@article{Read2000,
abstract = {There has been a considerable search for radical, amenable Banach algebras. Noncommutative examples were finally found by Volker Runde [R]; here we present the first commutative examples. Centrally placed within the construction, the reader may be pleased to notice a reprise of the undergraduate argument that shows that a normed space with totally bounded unit ball is finite-dimensional; we use the same idea (approximate the norm 1 vector x within distance η by a “good” vector $y_1$; then approximate $(x-y_1)/η$ within distance η by a “good” vector $y_2$, thus approximating x within distance $η^2$ by $y_1+η y_2$, and so on) to go from η=9/10 in Lemma 1.5 to arbitrarily small η in Lemma 2.1. This is not an arbitrary decision on the part of the author; it really is forced on him by the nature of the construction, see e.g. (6.1) for a place where η small at the start will not do.},
author = {Read, C.},
journal = {Studia Mathematica},
keywords = {adical; Banach algebra; amenable; nilpotent; radical},
language = {eng},
number = {3},
pages = {199-212},
title = {Commutative, radical amenable Banach algebras},
url = {http://eudml.org/doc/216764},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Read, C.
TI - Commutative, radical amenable Banach algebras
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 3
SP - 199
EP - 212
AB - There has been a considerable search for radical, amenable Banach algebras. Noncommutative examples were finally found by Volker Runde [R]; here we present the first commutative examples. Centrally placed within the construction, the reader may be pleased to notice a reprise of the undergraduate argument that shows that a normed space with totally bounded unit ball is finite-dimensional; we use the same idea (approximate the norm 1 vector x within distance η by a “good” vector $y_1$; then approximate $(x-y_1)/η$ within distance η by a “good” vector $y_2$, thus approximating x within distance $η^2$ by $y_1+η y_2$, and so on) to go from η=9/10 in Lemma 1.5 to arbitrarily small η in Lemma 2.1. This is not an arbitrary decision on the part of the author; it really is forced on him by the nature of the construction, see e.g. (6.1) for a place where η small at the start will not do.
LA - eng
KW - adical; Banach algebra; amenable; nilpotent; radical
UR - http://eudml.org/doc/216764
ER -

References

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  1. [BD] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973. Zbl0271.46039
  2. [DW] P. G. Dixon and G. A. Willis, Approximate identities in extensions of topologically nilpotent Banach algebras, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 45-52. Zbl0799.46060
  3. [G] G N. Grοnbæk, Amenability and weak amenability of tensor algebras and algebras of nuclear operators, J. Austral. Math. Soc. 51 (1991), 483-488. Zbl0758.46040
  4. [GJW] N. Grοnbæk, B. E. Johnson and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), 289-324. Zbl0806.46058
  5. [H] H U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. Zbl0529.46041
  6. [J] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
  7. [LRRW] R. J. Loy, C. J. Read, V. Runde and G. A. Willis, Amenable and weakly amenable Banach algebras with compact multiplication, J. Funct. Anal., to appear. Zbl0946.46041
  8. [R] V. Runde, The structure of contractible and amenable Banach algebras, in: E. Albrecht & M. Mathieu (eds.), Banach Algebras '97, de Gruyter, Berlin, 1998, 415-430. Zbl0927.46028

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