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

TL;DR

This paper establishes a criterion for standard form of von Neumann algebras using an unbounded intertwining operator, extends the BT-Theorem to show boundedness of certain operators, and explores automatic boundedness in algebraic morphisms and weights.

Contribution

It introduces a new criterion for standard form of von Neumann algebras and generalizes the BT-Theorem to prove automatic boundedness of intertwining operators.

Findings

Criterion for standard form via an intertwining operator.

Generalized BT-Theorem ensuring boundedness of certain operators.

Automatic boundedness results for algebraic morphisms and weights.

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 $\varphi\,$, whose left ideal $\mathfrak N_{\varphi}$ admits an algebraic complement in the GNS representation space $H_{\varphi}\,$, invariant under the canonical action of $M$.