Norms of basic elementary operators on algebras of regular operators

Abstract

The Banach algebra of all bounded linear operators on a Banach space is reasonably well understood, even though many unanswered questions about it remain. By contrast the space of all regular operators, even on a Dedekind complete Banach lattice, remains rather mysterious. We illustrate this by looking at basic elementary operators. These are operators, on an algebra of operators, of the form MA,B : T 7→ ATB for fixed operators A and B. It is difficult to imagine simpler operators. In the Banach space setting, it is a simple consequence of the Hahn-Banach theorem that ‖MA,B‖ = ‖A‖‖B‖. It is rather more difficult to compute the norm of a more general elementary operator, of the form ∑n k=1 MAk,Bk but several people in QUB have obtained significant results in that setting. By contrast even the norms of basic elementary operators on spaces of regular operators remain puzzling. I will give enough definitions so that the audience can make sense of the problem, and try to give some flavour of the methods used in proofs.