A twisted Hilbert space not isomorphic to its dual

TL;DR

This paper constructs the first example of a twisted Hilbert space not isomorphic to its dual, introduces a large family of such examples, and explores their properties, solving several open problems in the theory of twisted Hilbert spaces.

Contribution

It provides the first known twisted Hilbert space not isomorphic to its dual and demonstrates the existence of many incomparable such spaces, advancing the understanding of their structure.

Findings

Existence of a twisted Hilbert space not isomorphic to its dual

Large coneable family of incomparable twisted Hilbert spaces

Existence of quasilinear maps not isomorphic to Kalton centralizers

Abstract

We show: 1) The existence of the first twisted Hilbert space that is not isomorphic to its dual; this solves a problem posed by Cabello in [Nonlinear centralizers in homology, Math. Ann. 358 (2014), no. 3-4, 779-798]. 2) The existence of a large coneable family of relatively incomparable such examples, improving the coneable family obtained in [W.H. Corr\^{e}a, S. Dantas, D.L. Rodr\'iguez-Vidanes, Twisted Hilbert spaces defined by Lipschitz embeddings, Israel J. of Mathematics, to appear]. 3) The existence of quasilinear maps between Hilbert spaces not isomorphic to Kalton centralizers; which solves another question of Cabello. 4) The existence of a large family of mutually incomparable elements in the ordered set of twisted Hilbert exact sequences. This complements earlier results in [J.M.F. Castillo, W. Cuellar, V. Ferenczi, Y. Moreno, Complex structures on twisted Hilbert spaces, Israel J. Math. 222 (2017) 787-814] -- where it was proved that the ordered set did not have a first element -- and [F. Cabello S\'anchez, J.M.F. Castillo, W.H.G. Corr\^{e}a, V. Ferenczi, R. Garc\'ia, On the $Ext^2$-problem in Hilbert spaces, J. Funct. Anal. 280 (2021) 108863] -- where two incomparable elements were obtained.