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Lipschitz tensor product. (English) Zbl 1356.46015
Following the lead of 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 \(\mathrm{Lip}_0(X,E^*)\).
A 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.
There 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).
In 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.
The spaces of Lipschitz compact (finite-rank, approximable) operators from \(X\) to \(E^*\) are also described in terms of Lipschitz tensor products.

MSC:
46B28 Spaces of operators; tensor products; approximation properties
26A16 Lipschitz (Hölder) classes
46E15 Banach spaces of continuous, differentiable or analytic functions
47L20 Operator ideals
46M05 Tensor products in functional analysis
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