A Property in Vector-Valued Function Spaces

Abstract

This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur–Ulam property. We prove that a Banach space X enjoys this property if and only if C(K, X) enjoys this property. We also show the same result holds for $$L_\infty (\mu ,X)$$ and $$L_1(\mu ,X)$$ .