Nonlinear centralizers in homology II. The Schatten classes

  • Félix Cabello Sánchez

    Universidad de Extremadura, Badajoz, Spain
Nonlinear centralizers in homology II. The Schatten classes cover
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Abstract

An extension of by is a short exact sequence of quasi Banach modules and homomorphisms . When properly organized all these extensions constitute a linear space denoted by , where 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 , the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows:

In the first case, every extension splits and so . In the second case, every self-extension of arises (and gives rise) to a minimal extension of in the quasi Banach category, that is, a short exact sequence . In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence . 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.

Cite this article

Félix Cabello Sánchez, Nonlinear centralizers in homology II. The Schatten classes. Rev. Mat. Iberoam. 37 (2021), no. 6, pp. 2309–2346

DOI 10.4171/RMI/1265