Invariant means on double coset spaces (Q681546)

For a locally compact group \(G\) and a compact subgroup \(K\), the author introduces the definition of a \(K\)-invariant mean: it is a continuous positive linear functional \(M\) on the space \({\mathcal BC}(G//K)\), with \(M(1)=1\), which is invariant by the \(K\)-translation operators \(\tau_y\) (\(y\in G\)): \[ \tau_yf(x)=\int_K f(ykx)dk. \] (\({\mathcal BC}(G//K)\) is the space of continuous bounded functions on \(G\) which are \(K\)-biinvariant.) The pair \((G,K)\) is said to be amenable if there exists a \(K\)-invariant mean. The author proves that a Gelfand pair is amenable. As an application a stability property is proven for functions \(a\) on a commutative group \(H\) which are \(K\)-additive in the following sense: \[ {1\over | K|} \sum_{k\in K} a(x+k\cdot y)=a(x)+a(y), \] where \(K\) is a finite group of automorphisms of \(H\).