$π$-Properties, Uniformly Convexity and Uniform Ball Coverings Properties

We prove a sufficient criterion for closed subspaces of operator spaces containing the finite-rank operators to have the uniform ball-covering property. Let $F$ be a separable uniformly convex Banach space, and let $\Lambda_F>1$ be a constant determined by its modulus of convexity. If $F$ has the $\pi_\lambda$-property for some $1\leq \lambda < \Lambda_F$, then for every Banach space $E$ with separable dual, every closed subspace of $\mathcal{B}(E,F)$ containing $\mathcal{F}(E,F)$ has the UBCP. The proof uses a contraction estimate for near-metric finite-rank projections on uniformly convex spaces. We use this estimate to construct uniform ball coverings for the corresponding operator spaces. As applications, we obtain the UBCP for closed operator subspaces whose range spaces are vector-valued $L_p$-spaces, or separable uniformly convex $\mathcal{L}_{p,C+}$-spaces.