Journal of the Australian Mathematical Society

Research Article

Relative amenability and the non-amenability of B(l1)

C. J. Reada1

a1 Faculty of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom, e-mail:


In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras S1446788700014038_inline1 is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

(Received February 06 2004)

(Revised January 06 2005)

2000 Mathematics subject classification

  • primary 46J20;
  • 46H10;
  • secondary 46H20;
  • 13A05;
  • 13G05;
  • 46J05

Keywords and phrases

  • amenable;
  • Banach algebra;
  • operator algebra.