Commutative, radical amenable Banach algebras

• Autorzy: C. J. Read
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

adical; Banach algebra; amenable; nilpotent

Kategorie tematyczne

• 46H25: Normed modules and Banach modules, topological modules (if not placed in \SbjNo 13-XX or \SbjNo 16-XX)
• 46H20: Structure, classification of topological algebras
• 46J45: Radical Banach algebras
• 46B45: Banach sequence spaces
• 46B25: Classical Banach spaces in the general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 140
• Numer: 3
• Strony: 199-212

Daty

wydano

2000

otrzymano

1999-06-22

poprawiono

2000-02-04

Twórcy

autor

C. J. Read

• Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, United Kingdom

Bibliografia

• [BD] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973.
• [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.
• [G] G N. Grοnbæk, Amenability and weak amenability of tensor algebras and algebras of nuclear operators, J. Austral. Math. Soc. 51 (1991), 483-488.
• [GJW] N. Grοnbæk, B. E. Johnson and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), 289-324.
• [H] H U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319.
• [J] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
• [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.
• [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.