A note on lower bounds of norms of averaging operators (Q1581202)

Let \(S\) be an infinite closed subset of the compact space \(K\) and let \(K/S\) be the quotient space obtained by identifying \(S\) to a single point. Let \(\phi:K\rightarrow K/S\) be the quotient map. An averaging operator for \(\phi\) is a continuous linear operator \(u:C(K)\rightarrow C(K/S)\) such that \(u(f\circ\phi)=f\) for each \(f\in C(K/S).\) The authors show that if \(S\) contains \(n\) pairwise disjoint open subsets that are not separate (in a certain sense) in \(K,\) then every averaging operator for the quotient map has the norm greater or equal to \(2n+1.\) Examples of such operators are constructed.