# Commutative, radical amenable Banach algebras

Studia Mathematica (2000)

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

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topRead, 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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- [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
- [H] H U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. Zbl0529.46041
- [J] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). Zbl0256.18014
- [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
- [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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