Non-surjective coarse embeddings of continuous functions spaces (Q6073810)

Let \(D\) and \(E\) be Banach spaces. Let \(F: D \longrightarrow E\) be a map. For \(t \geq 0\), we define \[ \varphi_F(t) := \sup \{ | \|F(u) - F(v)\| - \|u - v\| |: \|u-v\| \leq t \ \mbox{or} \ \|F(u) - F(v)\| \leq t \} \] and we say that \(F\) is a coarse isometry if \(\varphi_F(t) = o(t)\) when \(t \rightarrow \infty\). In the present paper, the author is interested in the study of coarse isometries between spaces of continuous functions. More precisely, his main result is about the stability of almost surjective coarse embeddings \(F: C(X) \longrightarrow C(Y)\) between \(C(K)\)-spaces (see Theorem~5).