Completely bounded Banach-Mazur distance

Abstract

Analogous to the Banach-Mazur distance between Banach spaces, we study the completely bounded Banach-Mazur distance between operator spaces.In many cases of Banach spaces and Hilbert spaces we show that the infimum is attained when T is the identity map, and X, Y have the same base space. This provides a machinery to compute and estimate dcb(X, Y). Later, using symmetric norming functions we construct counterexamples to show that distinct infinite dimensional homogeneous operator spaces may have finite cb-distance, and that two homogeneous Hilbertian operator spaces may not coincide even if they coincide over all 2-dimensional subspaces.