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Seminar on Geometry of Banach Spaces Posts

2021.04.21 Bruno Braga, University of Virginia

Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

Abstract: While the nonlinear geometry of Banach spaces has been extensively studied (especially in the past few decades), the nonlinear geometry of its noncommutative counterpart, i.e., of operator spaces, has been neglected until very recently. In this talk, I will discuss some recent developments in this field. In particular, I will introduce the notion of almost complete Lipschitz embeddability between operator spaces and explain why this leads to a nontrivial nonlinear theory. For that, Lipschitz free spaces of operator spaces will play an important role. (This is a joint with with Javier Alejandro Chávez-Domínguez and Thomas Sinclair).

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2021.04.07 Paweł Wójcik, Pedagogical University, Kraków

When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth Banach space and a mapping that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediateextension of the former result to infinite dimensions, even under an extra hypothesis of uniform smoothness. Regrettably, this result refutes a claim concerning smooth spaces from a recent paper [Aequationes Math. (2020)].

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