On Coarse Isometries and Linear Isometries between Banach Spaces

Abstract

Let X,Y be two Banach spaces and f:X→Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a Lp(1p∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.

Keywords

• coarse isometry
• linear isometry
• <i>C</i>(<i>K</i>) space
• <i>L_p</i> space
• finite dimensional space
• Banach space