Projection from space of \emph{two-Lipschitz} operators onto the space of Bilinear maps
Arindam Mandal
TL;DR
This work studies the complementability of the bilinear map space inside two-Lipschitz mappings. It employs invariant means and Lipschitz-free spaces to construct a norm-one projection from $BLip_0(X,Y;E)$ onto $Blin(X,Y;E)$ when $E$ is dual, and derives isometric quotient descriptions when $E$ is injective. A key result is that $BLip_0(X,Y;E)/^{\diamondsuit}\mathcal{D}$ is isometrically isomorphic to $L(\mathcal{D},E)$ for suitable $\mathcal{D}$, linking nonlinear Lipschitz geometry to operator spaces via $F(X)$ and $F(Y)$. The paper also furnishes a necessary and sufficient condition for a two-Lipschitz map to be bilinear, and provides a concrete ℝ-example showing the identified duality and predual structure in a familiar setting such as $L^{\infty}(\mathbb{R}^2)/\text{span}\{1\}$.
Abstract
In this article, we establish the existence of a norm-one projection from the space of all \emph{two-Lipschitz} operators onto the space of all bounded bilinear operators under certain conditions on the corresponding codomain spaces, using the method of invariant means. We also show that, when the codomain is an injective Banach space, the quotient of the \emph{two-Lipschitz} operator space by the bounded bilinear space is isometrically isomorphic to a specific operator space, via vector-valued duality. We conclude by proving a necessary and sufficient condition for a \emph{two-Lipschitz} operator to be a bilinear map. As an application of the theory developed here, we present an alternative proof that $\bigslant{L^{\infty}(\mathbb{R}\times \mathbb{R})}{span\{\textbf{1}\}}$ is a dual space.