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

2015/05/19 by Fidaleo, Francesco, Zsidó, László
#46L10 #47A05 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1505.04910

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 \mathfrak N_φ admits an algebraic complement in the GNS representation space H_φ , invariant under the canonical action of M.

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