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 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(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows: ⎧ ⎪ if 0 < q < p ≤ ∞ or p = q = ∞, ⎨0 ExtB (S p , S q ) = Ext C (S 1 , C) if q = p is finite, ⎪ ⎩ if 0 < p < q ≤ ∞. Ext C (H) In the first case, every extension 0 −→ S q −→ Z −→ S p −→ 0 splits and so X = S q ⊕ 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 −→ C −→ M −→ S 1 −→ 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.