Two-sided Følner condition on amenable locally compact groups (Q1358383)

The author proves a two-sided analog of the Følner theorem that yields a combinatorial criterion for testing the amenability of a locally compact group \(G\). The following conditions (a) and (b) are equivalent to the amenability of \(G\), respectively: (a) For given \(\varepsilon >0\), \(\delta >0\), and a compact set \(K \subset G\), there exist Borel sets \(U \subset G\) and \(N \subset K\) such that \(0<|U|<\infty\), \(|N|<\delta\) \((|\cdot |\) is the left Haar measure) and any element \(x\in K\backslash N\) obeys the conditions \[ |U|^{-1} |xU \Delta U|<\varepsilon \quad \text{and} \quad |U |^{-1} |Ux \Delta U|<\varepsilon. \] (b) For given \(\varepsilon >0\) and any compact set \(K \subset G\), there exists a Borel set \(U\subset G\) such that \(0<|U|<\infty\) and every element \(x\in K\) obeys the conditions \[ |U|^{-1} |xU \Delta U|<\varepsilon \quad \text{and} \quad |U|^{-1} |Ux \Delta U|<\varepsilon. \] The author states that the above two-sided analog of the Følner condition finds application in harmonic analysis in the class of essentially bounded functions on amenable locally compact groups, on which all two-sided invariant means coincide.