A twisted Hilbert space not isomorphic to its dual

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.