Nigel Kalton's work in isometrical Banach space theory

This paper surveys some of the late Nigel Kalton's contributions to Banach space theory. The paper is written for the Nigel Kalton Memorial Website http://mathematics.missouri.edu/kalton/, which is scheduled to go online in summer 2011

Lipschitz spaces and M-ideals

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

The Daugavet equation for operators not fixing a copy of C(S)C(S)

We prove the norm identity $\|Id+T\| = 1+\|T\|$, which is known as the Daugavet equation, for operators $T$ on $C(S)$ not fixing a copy of $C(S)$, where $S$ is a compact metric space without isolated points

Remarks on rich subspaces of Banach spaces

We investigate rich subspaces of $L_1$ and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products.Comment: 12 page