A homological characterization of topological amenability (Q441104)

Let \(G\) be a finitely generated group acting on a compact Hausdorff space \(X\). Equip the space \(C(G,\ell^1(G))\) of continuous \(\ell^1(G)\)-valued functions on \(X\) with the sup-\(\ell^1\) norm NEWLINE\[NEWLINE\|\xi\| = \sup_{x\in X}\sum_{g\in G}|\xi(x)(g)|.NEWLINE\]NEWLINE The summation map induces a continuous map \(\sigma: C(G,\ell^1(G)) \to C(X)\) and the preimage \(N_0(G,X)=\sigma^{-1}(0)\cong C(X;\ell^1(G))\) and identify the space of constant functions on \(X\) with \(\mathbb R\). The authors define the standard module of the action of \(G\) on \(X\) as \(W_0(G,X) = \sigma^{-1}(\mathbb R) = N_0(G,X) + \mathbb R\) which is a \(G\)-module short exact extension NEWLINE\[NEWLINE0 \longrightarrow N_0(G,X) \overset{i}\longrightarrow W_0(G,X) \overset{\sigma}\longrightarrow \mathbb R \longrightarrow 0 \,.NEWLINE\]NEWLINE The authors define the uniformly finite homology of the action as the group homology NEWLINE\[NEWLINEH^{uf}_n(G\rightarrow X) = H_n(G,W_0(G,X)^*).NEWLINE\]NEWLINE The main result of the paper is the theorem that the action of \(G\) on \(X\) is topologically amenable if and only if the fundamental class \([G\curvearrowright X]\) is non-zero in \(H^{uf}_n(G\to X)\).