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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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