Weak amenability components of \(L_1(G)\)-modules, amenable groups, and an ergodic theorem (Q1582815)

Let \(G\) be a locally compact group with fixed left Haar measure and let \(M\) be a left Banach \(L_1(G)\)-module. Then the dual \(M^*\) is also a left Banach \(L_1(G)\)-module defined as usual. The two bilinear operations are defined by \[ M\times M^*\to L_\infty(G), (x,\xi)\mapsto x\circ_1 \xi,\text{ where }\langle \varphi,x\circ_1 \xi\rangle= \langle\varphi^* \cdot x,\xi \rangle ; \] \[ M^*\times L_\infty (G)^*\to M^*,(\xi,\Phi) \mapsto\xi \circ_2\Phi, \text{ where }\langle x, \xi\circ_2 \Phi\rangle= \langle x\circ_1 \xi,\Phi \rangle \] for \(x\in M\), \(\xi\in M^*\), \(\varphi\in L_1(G)\), and \(\Phi\in L_\infty (G)^*\). \(M\) is said to be amenable if the set \(X=\{x\in M:\inf\{ \|\varphi \cdot x\|: \varphi\in P(G)\}= 0\}\) is closed with respect to addition, where \(P(G)= \{\varphi\in L_1(G): \varphi\geq 0,\|\varphi\|_1 =1\}\). For an arbitrary left Banach \(L_1(G)\)-module \(M\), there is a maximal amenable submodule, which is denoted by \(A(M)\) and is called the amenability component of \(M\). The weak \((\sim\)-weak, respectively) amenability component \(W(M^*)\) \((\widetilde W(M^*)\), respectively) of the module \(M^*\) is the set of all \(\xi\in M^*\) such that \(\xi \circ_1 x\in A(L_\infty (G))\) \((x\circ_1 \xi\in A(L_\infty (G))\), respectively) for all \(x\in M\). The author proves that an element \(\xi\) in \(W(M^*)\) can be characterized by the condition \((\varphi \circ\xi) \circ m=\xi \circ m\) for all \(\varphi\in P(G)\) and all topologically left invariant means \(m\) on \(A(L_\infty (G))^\sim\), or by the condition \(w^*-\lim_\alpha (\varphi_\alpha \cdot(\varphi \cdot\xi) -\varphi_\alpha \cdot\xi)=0 \) for all \(\varphi\in P(G)\) and all nets \(\{\varphi_\alpha\}\) in \(P(G)\) such that \(w^*- \lim_\alpha \varphi_\alpha\) is a topological left invariant mean on \(A(L_\infty (G))^\sim\). There is also a similar characterization for the elements of \(\widetilde W(M^*)\) in this paper. A mean ergodic theorem for locally compact groups is formulated. The module \(M^*\) is said to be weakly amenable \((\sim\)-weakly amenable, respectively) if \(M^*=W(M^*)\) \((M^*= \widetilde W(M^*)\), respectively). It is proved that the following conditions are equivalent: (i) \(G\) is amenable; (ii) the module \(M^*\) is weakly amenable for all left Banach \(L_1(G)\)-modules \(M\); (iii) the module \(M^*\) is \(\sim\)-weakly amenable for all left Banach \(L_1(G)\)-modules \(M\); (iv) \(W(L_\infty(G)) =L_\infty(G)\); (v) \(\widetilde W(L_\infty (G)) =L_\infty (G)\). The author also characterizes amenable locally compact groups by the condition \(W({\mathcal B}(X^*))= {\mathcal B}(X^*)\) or \(\widetilde W({\mathcal B}(X^*))= {\mathcal B}(X^*)\) for some Banach spaces \(X\) of functions on \(G\).