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 $\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, 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 $\ell^{\infty}$.Comment: Includes 4 figure