[Paper Review] A decomposition theorem in II_1-factors

Kenneth J. Dykema, Fedor Sukochev+1 more|arXiv (Cornell University)|Feb 5, 2013

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper establishes a decomposition theorem for operators in diffuse, finite von Neumann algebras, particularly II₁-factors, proving that any operator T can be uniquely written as the sum of a normal operator N and an s.o.t.-quasinilpotent operator Q. The key contribution is a non-commutative analogue of Schur's triangularization and Ringrose's compact operator decomposition, using Haagerup-Schultz projections and Brown measure to ensure N and T have identical spectral distributions while Q satisfies limₙ‖((Q*)ⁿQⁿ)¹/²ⁿ‖_∞ = 0.

ABSTRACT

Building on results of Haagerup and Schultz, we decompose an arbitrary operator in a diffuse, finite von Neumann algebra into the sum of a normal operator and an s.o.t.-quasinilpotent operator. We also prove an analogue of Weyl's inequality relating eigenvalues and singular values for operators in a diffuse, finite von Neumann algebra.

Motivation & Objective

• To extend Schur’s and Ringrose’s decomposition theorems to the setting of diffuse, finite von Neumann algebras and II₁-factors.
• To establish a non-commutative analogue of the spectral decomposition of matrices and compact operators in terms of normal and quasinilpotent components.
• To prove that in a diffuse, finite von Neumann algebra, every operator T decomposes uniquely as T = N + Q, where N is normal and Q is s.o.t.-quasinilpotent.
• To establish a Weyl-type inequality for eigenvalues and singular values in the II₁-factor setting using logarithmic submajorization.

Proposed method

• Utilizes Haagerup-Schultz projections associated with the Brown measure of an operator T in a II₁-factor to construct spectral projections for Borel sets in ℂ.
• Defines the normal operator N as the sum of rank-one operators T(p_λ - p_{λ-0}) over jump points of the spectral projection net, ensuring N is normal and has the same Brown measure as T.
• Constructs the remainder operator Q = T - N and proves it is s.o.t.-quasinilpotent by showing limₙ‖((Q*)ⁿQⁿ)¹/²ⁿ‖_∞ = 0, which is equivalent to the Brown measure of Q being supported at {0}.
• Applies logarithmic submajorization (≺≺_log) to compare singular values of T and N, using the trace inequality τ(log₊(|B|/t)) ≤ τ(log₊(|A|/t)) for B ≺≺_log A.
• Uses convexity and functional calculus to derive τ(Φ(|N|)) ≤ τ(Φ(|T|)) for all increasing Φ with Φ∘exp convex, establishing a Weyl-type inequality.
• Employs the monotone convergence principle and approximation via |N| + 1/n to pass from regularized inequalities to the final trace inequality.

Experimental results

Research questions

• RQ1 Can the decomposition of matrices and compact operators into normal and quasinilpotent parts be extended to operators in diffuse, finite von Neumann algebras?
• RQ2 Does every operator in a II₁-factor admit a decomposition into a normal operator and an s.o.t.-quasinilpotent operator with matching Brown measure?
• RQ3 To what extent do eigenvalue and singular value inequalities (Weyl-type) hold in the non-tracial, non-commutative setting of II₁-factors?
• RQ4 How can Haagerup-Schultz projections be used to construct such a decomposition in a canonical, hyperinvariant way?
• RQ5 What is the precise relationship between the Brown measure of T and the spectral distribution of the normal part N in the decomposition T = N + Q?

Key findings

• Every operator T in a diffuse, finite von Neumann algebra with a normal, faithful tracial state τ decomposes uniquely as T = N + Q, where N is normal and Q is s.o.t.-quasinilpotent.
• The Brown measure of N equals the Brown measure of T, ensuring spectral invariance under the decomposition.
• The remainder operator Q satisfies limₙ‖((Q*)ⁿQⁿ)¹/²ⁿ‖_∞ = 0, which characterizes s.o.t.-quasinilpotency in finite von Neumann algebras.
• The normal part N satisfies N ≺≺_log T, meaning τ(log₊(|N|/t)) ≤ τ(log₊(|T|/t)) for all t > 0.
• For every increasing function Φ on [0, ∞) such that Φ∘exp is convex, the inequality τ(Φ(|N|)) ≤ τ(Φ(|T|)) holds, generalizing Weyl’s inequality to the II₁-factor setting.
• The decomposition is constructed via Haagerup-Schultz projections associated with the Brown measure, ensuring hyperinvariance and uniqueness.