On Qian's problem for $\mathcal{L}_{\infty}$-spaces

Abstract

In this paper we devote to study Qian's problem for $\mathcal{L}_{\infty}$-spaces. Firstly, a positive answer to Qian's problem for $C(K)$-spaces is given by the assumption that $K$ has the C$\check{e}$ch-Stone property. Secondly, we obtain quantitative characterizations of separably injective spaces that turn out to give a positive answer to Qian's problem of 1995 in the setting of separable universality. Thirdly, we prove a sharpen quantitative and generalized Sobczyk theorem, which gives sharpen constants ($\alpha,\gamma$) for Qian's Problem. Finally, we give a more generalized Figiel theorem for $\mathcal{L}_{\infty}$-spaces.