[Paper Review] Quantitative BT-Theorem and automatic continuity for standard von Neumann algebras

Francesco Fidaleo, László Zsidó|arXiv (Cornell University)|May 19, 2015

Advanced Topics in Algebra23 references4 citations

TL;DR

This paper establishes a quantitative version of the Borchers-Takesaki (BT) theorem for standard von Neumann algebras, proving that any everywhere defined, invertible, antilinear operator intertwining a von Neumann algebra with its commutant and acting as the *-operation on the center must be automatically bounded. This result leads to a general automatic continuity theorem for intertwining operators, with applications to morphisms of von Neumann algebras and the structure of W*-algebras with normal, semi-finite, faithful weights possessing invariant GNS complements.

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 $M$.

Motivation & Objective

• To establish a general criterion for a von Neumann algebra to be in standard form using an antilinear intertwining operator.
• To prove that such an intertwining operator, though not a priori bounded, must be automatically bounded.
• To generalize the classical BT-Theorem to derive automatic continuity for operators intertwining the identity map with bounded real linear operator-valued maps.
• To apply the automatic continuity result to the boundedness of linear maps implementing algebraic morphisms from von Neumann algebras to Banach algebras.
• To analyze the structure of W*-algebras equipped with a normal, semi-finite, faithful weight whose GNS left ideal admits an M-invariant algebraic complement.

Proposed method

• Introduce a criterion for standardness of a von Neumann algebra M via the existence of a bijective antilinear operator T satisfying TMT⁻¹ = M′ and TzT⁻¹ = z* for all z ∈ Z(M).
• Prove that such an operator T must be bounded using a generalized version of the BT-Theorem, which extends the classical result to a broader class of intertwining operators.
• Establish a general automatic continuity theorem (Theorem 5.1) for operators intertwining the identity map on a von Neumann algebra with a bounded, real linear, operator-valued map.
• Use the generalized BT-Theorem to show that any linear operator implementing an algebraic morphism from a von Neumann algebra to a Banach algebra is automatically bounded.
• Analyze the GNS representation associated with a normal, semi-finite, faithful weight φ on a W*-algebra M, focusing on the existence of an M-invariant algebraic complement to the left ideal Nφ in Hφ.
• Construct explicit examples of unbounded linear and antilinear involutions on infinite-dimensional Hilbert spaces using Hamel bases to illustrate the necessity of boundedness assumptions.

Experimental results

Research questions

• RQ1 Under what conditions is a von Neumann algebra M on a Hilbert space H standard, i.e., spatially isomorphic to its commutant via an antilinear conjugation?
• RQ2 Can an antilinear, everywhere defined, invertible operator T intertwining M with M′ and acting as * on Z(M) be necessarily bounded, even if not assumed a priori?
• RQ3 What general conditions ensure the automatic boundedness of operators intertwining the identity map on a von Neumann algebra with a bounded, real linear, operator-valued map?
• RQ4 How does the existence of an M-invariant algebraic complement to the GNS left ideal Nφ affect the structure of a W*-algebra M equipped with a normal, semi-finite, faithful weight φ?
• RQ5 Are there unbounded linear or antilinear involutions on infinite-dimensional Hilbert spaces, and if so, how can they be constructed?

Key findings

• A von Neumann algebra M is standard if and only if there exists a bijective antilinear operator T: H → H such that TMT⁻¹ = M′ and TzT⁻¹ = z* for all z ∈ Z(M).
• Any such intertwining operator T, even if not assumed bounded, is necessarily bounded, establishing automatic continuity for this class of operators.
• The generalized BT-Theorem (Theorem 4.1) ensures that any operator intertwining the identity on a von Neumann algebra with a bounded, real linear, operator-valued map is automatically bounded.
• The automatic continuity result (Theorem 5.1) implies that any linear operator implementing an algebraic morphism from a von Neumann algebra onto a Banach algebra is necessarily bounded.
• For a W*-algebra M with a normal, semi-finite, faithful weight φ, if the GNS left ideal Nφ admits an M-invariant algebraic complement in Hφ, then M is finite-dimensional.
• There exist unbounded linear and antilinear involutions on infinite-dimensional Hilbert spaces, constructed via Hamel bases with finitely many continuous coordinate functionals.