Details

Autor(en) / Beteiligte

• Sanchez, Felix Cabello

Titel

Nonlinear centralizers in homology II. The Schatten classes

Ist Teil von

• Revista matemática iberoamericana, 2021-08, Vol.37 (6), p.2309-2346

Ort / Verlag

European Mathematical Society Publishing House

Erscheinungsjahr

2021

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0 [right arrow] Y [right arrow] Z [right arrow] X [right arrow] 0. When properly organized all these extensions constitute a linear space denoted by [Ext.sub.B] (X, Y), where B is the underlying (Banach) algebra. In this paper we "compute" the spaces of extensions for the Schatten classes when they are regarded in its natural (left) module structure over B = B(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows: [Please download the PDF to view the mathematical expression] In the first case, every extension 0 [right arrow] [S.sup.q] [right arrow] Z [right arrow] [S.sup.p] [right arrow] 0 splits and so [Please download the PDF to view the mathematical expression]. In the second case, every self-extension of [S.sup.p] arises (and gives rise) to a minimal extension of [S.sup.1] in the quasi Banach category, that is, a short exact sequence 0 [right arrow] C [right arrow] M [right arrow] [S.sup.1] [right arrow] 0. In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence 0 [right arrow] H [right arrow] T [right arrow] H [right arrow] 0. Thus, the subject of the paper is closely connected to the early "three-space" problems studied (and solved) in the seventies by Enflo, Lindenstrauss, Pisier, Kalton, Peck, Ribe, Roberts, and others.