Rosenthal sets for Banach-valued functions
Let \(G\) be a compact metric abelian group with doubling normalized Haar measure \(\mu\). Let \(X\) be a Banach space not containing \(c_0\). We show that if \(H\) is a dense subgroup of \(G\), if \(F \in L^\infty (G,X)\), and if all the differences \(\Delta_h F(t) = F(t + h) - (t)\), \(h \in H\), are \(\mu\)-a.e. equal to continuous functions on \(G\) then \(F\) also is \(\mu\)-a.e. equal to a continuous function on \(G\). -- We apply this result in the following problem of harmonic analysis: let \(G\), \(X\), \(H\) be as above, \(\Gamma\) be the dual group of \(G\), and suppose \(H\) is countable. If we denote by \(\widehat H\) the dual group of \(H\), where \(H\) is given the discrete topology, then we can identify \(\Gamma\) with a dense subgroup of \(\widehat H\). Let \(\Lambda\) be a subset of \(\Gamma\) with countable closure in \(\widehat H\). F. Piquard has proved that under such hypothesis we have \(L^\infty_\Lambda (G) = C_\Lambda (G)\). Here we generalize this result to Banach-valued functions by showing that if \(X\) is a Banach space not containing \(c_0\) and if \(\Lambda\) is as above then \(L^\infty_\Lambda (G,X) = C_\Lambda (G,X)\). We show that this happens also in the case where \(\Lambda\) is any Rosenthal set and \(X\) is any Banach space with the Schur property.
scientific article; zbMATH DE number 938562
scientific article