Amenable \(L_1(G)\)-modules, averaging functions, and spaces with mixed norm (Q1293609)

Let \(G\) be a locally compact group. A function space \(L_\rho (G)\) on \(G\) with a functional norm \(\rho\) is called function \(G\)-module if the space is complete and left translations define isometries of the space \(L_\rho (G)\). The main problem studied in the paper is the amenability of the above modules. Averaging functions are used for the characterizations. Let \(P(G)= \{\psi\in L_1(G): \psi\geq 0\), \(\|\psi \|_1=1\}\). A function \(f\in L_\infty(G)\) is called topological left averaging with constant \(c\) if the closure of the set \(P(G)*f\) contains the function \(c\cdot 1_G\), \((1_G(x)\equiv 1)\). The author gives different equivalent characterizations of topological left averaging functions from maximal amenable submodules of \(L_\infty(G)\). The result is used for the characterization of the amenability of the function \(G\)-modules. In particular, it is proved that if the norm \(\rho\) is absolutely continuous then \(L_\rho(G)\) is amenable if and only if \(G\) is amenable or a submodule \({\mathcal L}_\rho\) of \(L_\infty(G)\) generated by the functions \(x\circ y\), \(x\in L_\rho(G)\) and \(y\in L_\rho^*(G)\), is contained in the set \[ \bigl\{f \in L_\infty (G):\varphi *|f|\text{ is topological left averaging for any }\varphi\in P(G)\bigr\}, \] where the functions \(x\circ y\) are defined by \(\langle\varphi,x\circ y\rangle=\langle\varphi^*x,y\rangle\) \((\varphi\in L_1 (G))\). Let \(p=(p_1, \dots,p_n)\). If the norm \(\rho\) is a mixed \(L_p\)-norm with respect to a finite increasing family of normal subgroups \(\{e\}=N_0\subset N_1 \subset\cdots \subset N_n=G\), then \(L_p(G)\) is amenable if and only if \(G\) is amenable or there is \(i\) such that the factor group \(N_{i+1}/N_i\) is not compact and \(p_i>1\).