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 $arphi,$, whose left ideal $mathfrak N_arphi$ admits an algebraic complement in the GNS representation space $H_arphi,$, invariant under the canonical action of $M$.
Fidaleo, F., Zsido, L. (2016). Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras. ADVANCES IN MATHEMATICS, 289, 1236-1260 [10.1016/j.aim.2015.07.039].
Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras
FIDALEO, FRANCESCO;ZSIDO, LASZLO
2016-05-19
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 $arphi,$, whose left ideal $mathfrak N_arphi$ admits an algebraic complement in the GNS representation space $H_arphi,$, invariant under the canonical action of $M$.File | Dimensione | Formato | |
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