Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $L\Lambda \subset L\Gamma$ arising from amenable subgroups $\Lambda$ of discrete countable groups $\Gamma$. The criterion is expressed in terms of $\Lambda$-invariant measures on some compact $\Gamma$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C*-algebra.