Lipschitz tensor product (Q2800619)

Following the lead of \textit{R. Schatten} [A theory of cross-spaces. Princeton, N. J.: Princeton University Press (1950; Zbl 0041.43502)], the authors introduce the ``Lipschitz tensor product'' of a pointed metric space \(X\) and a Banach space \(E\). For every \(x \in X\) and \(e \in E\), the Lipschitz tensor product \(\delta_{(x,0)} \boxtimes e\) is defined by \((\delta_{(x,0)} \boxtimes e)(f) = \langle f(x), e \rangle\) for \(f \in \mathrm{Lip}_0(X,E^*)\). The Lipschitz tensor product \(X \boxtimes E\) is the linear span of all such simple tensors in the algebraic dual of \(\mathrm{Lip}_0(X,E^*)\).NEWLINENEWLINEA norm \(\alpha\) on \(X \boxtimes E\) is a Lipschitz cross-norm if \(\alpha(\delta_{(x,y)} \boxtimes e) = d(x,y)\|e\|\). To ensure the good behavior of a norm on \(X \boxtimes E\) with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on \(X \boxtimes E\) is defined.NEWLINENEWLINEThere is a least dualizable Lipschitz cross-norm \(\varepsilon\) and a greatest Lipschitz cross-norm \(\pi\) on \(X \boxtimes E\). Both \(\varepsilon\) and \(\pi\) are uniform. Dualizable Lipschitz cross-norms \(\alpha\) on \(X \boxtimes E\) are characterized by satisfying the relation \(\varepsilon \leq \alpha \leq \pi\). The norm \(\varepsilon\) is called injective and it is shown that it respects injections (Theorem~5.6), and \(\pi\) is called projective since it is shown to respect projections (Proposition~6.12).NEWLINENEWLINEIn addition, the Lipschitz injective (projective) norm on \(X \boxtimes E\) can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over \(X\) and \(E\), but this identification does not hold for the Lipschitz \(2\)-nuclear norm and the corresponding Banach-space tensor norm.NEWLINENEWLINEThe spaces of Lipschitz compact (finite-rank, approximable) operators from \(X\) to \(E^*\) are also described in terms of Lipschitz tensor products.