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Commutative, radical amenable Banach algebras

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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.
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  • Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, United Kingdom, cr25@cam.ac.uk
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.
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bwmeta1.element.bwnjournal-article-smv140i3p199bwm
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