Lipschitz spaces and M-ideals

Abstract

For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\Vert f{\Vert }_L=\max (\Vert f{\Vert }_{\infty },L(f))$, where $L(f)$ is the Lipschitz constant of $f$. The closed subspace $\lip(K)$ of $\Lip(K)$ contains all elements of $\Lip(K)$ satisfying the $\lip$-condition ${\lim }_{0<d(x,y)\to 0}|f(x)-f(y)|/d(x,y)=0$. For $K=([0,1],|\cdot {|}^{\alpha })$, $0<\alpha <1$, we prove that $\lip(K)$ is a proper $M$-ideal in a certain subspace of $\Lip(K)$ containing a copy of ${\ell }^{\infty }$.

Includes 4 figures