Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras

Abstract

We prove a general criterion for a von Neumann algebra M in order to be in standard form. It is formulated in terms of an everywhere defined, invertible, antilinear, a priori not necessarily bounded operator, intertwining M with its commutant M′ and acting as the ⁎-operation on the centre. We also prove a generalized version of the BT-Theorem which enables us to see that such an intertwiner must be necessarily bounded. It is shown that this extension of the BT-Theorem leads to the automatic boundedness of quite general operators which intertwine the identity map of a von Neumann algebra with a general bounded, real-linear, operator-valued map. We apply the last result to the automatic boundedness of linear operators implementing algebraic morphisms of a von Neumann algebra onto some Banach algebra, and to the structure of a W⁎-algebra M endowed with a normal, semi-finite, faithful weight φ, whose left ideal Nφ admits an algebraic complement in the GNS representation space Hφ, invariant under the canonical action of M.