Weak *-closed complemented invariant subspaces of \(L_{\infty}(G)\) and amenable locally compact groups
Let G be a locally compact group admitting the modular function \(\Delta\). If \(g\in G\) and \(\Phi \in L_ 1(G)\), \(f\in L_{\infty}(G)\), consider \(l_ gf(x)=f(g^{-1}x)\), \(\tau_ g(\Phi)(x)=\Phi (g^{-1}xg) \Delta (g)\), \(x\in G\). A left Banach G-module X is a Banach space X that constitutes a left G-module and satisfies the following conditions: (i) \(\| s\cdot x\| \leq \| x\|\) for all \(x\in X\), \(s\in G\); (ii) for every \(x\in X\), the mapping \(G\to X\), \(s\mapsto s\cdot x\) is continuous. The action may be carried over to \(X^*\). The weak *-closed left translation invariant subspace of \(L_{\infty}(G)\) is called invariantly complemented if it admits a left translation invariant, closed complement, or equivalently, X is the range of a continuous projection on \(L_{\infty}(G)\) commuting with left translations. The authors rely on a result due to the second author and \textit{H. Rindler} [Lect. Notes Math. 1064, 368-378 (1984; Zbl 0573.43003)]: If G is an amenable group, then \(L_ 1(G)\) admits a net \((u_{\alpha})\), bounded by 1 with respect to the \(\| \|_ 1\)-norm, such that lim \(\| \tau_ gu_{\alpha}-u_{\alpha}\|_ 1=0\) whenever \(g\in G\). The main statement asserts that the locally compact group G is amenable if and only if for every non-degenerate left Banach G-module X and every weak *-closed G-invariant complemented subspace L of X, there exists a projection Q of \(X^*\) onto L such that \(Q(f\cdot s)=Q(f)\cdot s\) whenever \(s\in G\), \(f\in X^*.\) An extension of a property obtained by the first author [Ill. J. Math. 26, 226-235 (1982; Zbl 0486.43002)] is established. Consider the von Neumann algebra VN(G) generated by \(\{l_ g:\) \(g\in G\}\) on \(L_ 2(G)\) and the Fourier algebra \(A(G)\), predual of \(VN(G)\). If M is an \(A(G)\)- invariant \(W^*\)-subalgebra of VN(G) such that \(\{g\in G:\) \(l_ g\in M\}\) is a normal subgroup of G, then there exits a continuous projection P of VN(G) onto M such that \(P(\Phi \cdot x)=\Phi \cdot P(x)\) whenever \(\Phi\in A(G)\) and \(x\in VN(G)\). In particular, M admits an invariant closed complement.
scientific article; zbMATH DE number 3948963