J. Aust. Math. Soc.  80 (2006), 317-333
Relative amenability and the non-amenability of $B(l^1)$

C. J. Read
  Faculty of Mathematics
  University of Leeds
  Leeds LS2 9JT
  United Kingdom

In this paper we begin with a short, direct proof that the Banach algebra $B(l^1)$ 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 $\bigoplus_{n=1}^\infty B(l^p_n)$ is not amenable for $1\le p\le\infty$, $p\ne 2$. 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 $X=l^2$) and we introduce the notion of 'relative amenability' of these algebras.
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