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I search the reference for the proof of the following theorem:

Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable.

The reference

Dales/Ghahramani/Helemskii. The amenability of measure algebras. J. London Math. Soc. (2) 66 (2002), no. 1, 213–226.

is often cited. But it seems to me that the authors only proved the following assertion: $G$ is discrete and amenable if and only if $M(G)$ is amenable.

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up vote 5 down vote accepted

Here is the text of an email I got from a friend who is expert on such matters:

"This is indeed the usual reference.

If $G$ is discrete, them $M(G)$ is just the group algebra. This is known to be weakly amenable by a result of Barry Johnson from 1988.

MR0974315 (90a:46122) Johnson, B. E. [Johnson, Barry Edward] (4-NWCT)

Derivations from $L^1(G)$ into $L^1(G)$ and $L^\infty(G)$.

Harmonic analysis (Luxembourg, 1987), 191–198, Lecture Notes in Math., 1359, Springer, Berlin, 1988.

As far as the converse is concerned, the proof of the quoted result builds a non-zero point derivation on $M(G)$ for any non-discrete $G$. This is well known to imply non-weak amenability."

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