Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras
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 cent...
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Zusammenfassung: | 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 $M$. |
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DOI: | 10.48550/arxiv.1505.04910 |