Nonlinear centralizers in homology II. The Schatten classes

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 ${\mathrm{Ext}}_B(X,Y)$, 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 $B=B(\mathcal{H})$, the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows:

In the first case, every extension $0⟶S^q⟶Z⟶S^p⟶0$ splits and so $X=S^q\oplus S^p$. In the second case, every self-extension of $S^p$ arises (and gives rise) to a minimal extension of $S^1$ in the quasi Banach category, that is, a short exact sequence $0⟶\mathbb{C}⟶M⟶S^1⟶0$. In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence $0⟶\mathcal{H}⟶T⟶\mathcal{H}⟶0$. 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.