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Nonlinear centralizers in homology II. The Schatten classes.
- Source :
-
Revista Mathematica Iberoamericana . 2021, Vol. 37 Issue 6, p2309-2346. 38p. - Publication Year :
- 2021
-
Abstract
- An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0 -→ Y -→ Z -→ X -→ 0. When properly organized all these extensions constitute a linear space denoted by ExtB(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: ... In the first case, every extension 0 -→ Sq -→ Z -→ Sp -→ 0 splits and so X = Sq -Sp. In the second case, every self-extension of Sp arises (and gives rise) to a minimal extension of S1 in the quasi Banach category, that is, a short exact sequence 0 -→ C -→ M -→ S1 -→ 0. In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence 0 -→ H -→ T -→ H -→ 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *VECTOR spaces
*HILBERT space
*OPERATOR algebras
*HOMOMORPHISMS
*ALGEBRA
Subjects
Details
- Language :
- English
- ISSN :
- 02132230
- Volume :
- 37
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Revista Mathematica Iberoamericana
- Publication Type :
- Academic Journal
- Accession number :
- 152338210
- Full Text :
- https://doi.org/10.4171/rmi/1265