Linearization of Lipschitz framings for Banach spaces

Abstract

Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every $\mathrm{Lip}(X,Y)$-valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is $(N,2^N)$, and how nonlinear sampling naturally induces a Lipschitz framing.