[Paper Review] Approximate double commutants in von Neumann algebras

Don Hadwin|arXiv (Cornell University)|Aug 25, 2011

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper establishes that every commutative C*-subalgebra of a centrally prime C*-algebra equals its relative approximate double commutant, generalizing von Neumann's double commutant theorem to approximate settings. The key result shows that in a von Neumann algebra, the approximate double commutant of a commutative C*-subalgebra coincides with the C*-algebra it generates, with a related distance formula for hyperreflexivity.

ABSTRACT

Richard Kadison showed that not every commutative von Neumann subalgebra of a factor von Neumann algebra is equal to its relative double commutant. We prove that every commutative C*-subalgebra of a centrally prime C*-algebra $B$ equals its relative approximate double commutant. If $B$ is a von Neumann algebra, there is a related distance formula.

Motivation & Objective

• To extend von Neumann's double commutator theorem to approximate versions in C*- and von Neumann algebras.
• To investigate whether commutative C*-subalgebras in von Neumann algebras equal their relative approximate double commutant.
• To establish a distance formula for the approximate double commutant in von Neumann algebras, linking it to hyperreflexivity.
• To explore conditions under which the approximate double commutant equals the C*-algebra generated by the subalgebra.
• To examine the role of central projections and factor structures in determining the structure of approximate double commutants.

Proposed method

• Defines the relative approximate double commutant of a C*-subalgebra 𝒮 in a unital C*-algebra 𝒷 as the set of operators T in 𝒷 such that ‖AλT − TAλ‖ → 0 for all bounded nets {Aλ} in 𝒷 that asymptotically commute with every S ∈ 𝒮.
• Uses nets of projections and unitaries to characterize the approximate double commutant, showing it remains equal to C*(𝒮) even under such restrictions.
• Applies a lifting argument via ultraproducts of C*-algebras to reduce the problem to finite-dimensional algebras, leveraging the structure of central projections.
• Proves that if 𝒷 is centrally prime, then appr(𝒜,𝒷)′′ = C*(𝒜 ∪ ℤ(𝒷)), where ℤ(𝒷) is the center of 𝒷.
• Establishes that for norm-separable subsets 𝒮 of a C*-algebra 𝒷, appr(𝒮,𝒷)′′ = (𝒮,𝒷)′′, showing the approximate and standard double commutants coincide under separability.
• Uses the existence of partial isometries satisfying AV = VB = V under certain positivity and orthogonality conditions to construct approximating nets.

Experimental results

Research questions

• RQ1 Does every commutative C*-subalgebra of a centrally prime C*-algebra equal its relative approximate double commutant?
• RQ2 Is there a uniform distance formula relating the distance of an operator to a von Neumann subalgebra and the norm of its commutator with projections in the commutant?
• RQ3 Can the approximate double commutant theorem be extended to non-self-adjoint or non-commutative C*-subalgebras in factor von Neumann algebras?
• RQ4 Under what conditions on a C*-algebra 𝒷 does the approximate double commutant of a C*-subalgebra equal the C*-algebra it generates?
• RQ5 Does the existence of partial isometries satisfying AV = VB = V for positive elements A, B, X, Y with AX = X, BY = Y, AB = 0, and X𝒷Y ≠ {0} imply stronger structural properties in C*-algebras?

Key findings

• Every commutative C*-subalgebra 𝒜 of a centrally prime C*-algebra 𝒷 satisfies appr(𝒜,𝒷)′′ = C*(𝒜 ∪ ℤ(𝒷)).
• In a von Neumann algebra 𝒷, if 𝒜 is a commutative C*-subalgebra, then appr(𝒜,𝒷)′′ = C*(𝒜), meaning the approximate double commutant recovers the C*-algebra generated by 𝒜.
• For norm-separable subsets 𝒮 of a C*-algebra 𝒷, appr(𝒮,𝒷)′′ = (𝒮,𝒷)′′, showing that the approximate and standard double commutants coincide under separability.
• The paper establishes a distance formula: dist(T,𝒜) ≤ 29 limλ ‖TPλ − PλT‖ for a net of projections {Pλ} in 𝒜′, proving approximate hyperreflexivity for C*-subalgebras.
• If 𝒷 is a factor von Neumann algebra, then appr(𝒜,𝒷)′′ = C*(𝒜) holds for any commutative C*-subalgebra 𝒜 of 𝒷.
• The existence of partial isometries V satisfying AV = VB = V under positivity and orthogonality conditions is a key technical tool for constructing approximating nets in the proof.