[Paper Review] Additive derivations on algebras of measurable operators

Shavkat Ayupov, Karimbergen Kudaybergenov|ArXiv.org|Aug 9, 2009

Advanced Operator Algebra Research10 references19 citations

TL;DR

This paper introduces the central extension $\mathrm{mix}(M)$ of a von Neumann algebra $M$ and proves that every additive derivation on $\mathrm{mix}(M)$ is inner when $M$ is properly infinite. This result implies that all additive derivations on the algebra $\mathrm{LS}(M)$ are inner for type I$_\infty$ or type III von Neumann algebras, generalizing previous results on inner derivations in noncommutative measure theory.

ABSTRACT

Given a von Neumann algebra $M$ we introduce so called central extension $mix(M)$ of $M$. We show that $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, all additive derivations are inner derivations.

Motivation & Objective

• To investigate the structure of additive derivations on algebras of measurable operators affiliated with von Neumann algebras.
• To address the conjecture that non-inner derivations arise primarily in commutative settings, by studying noncommutative cases.
• To introduce and analyze the central extension $\mathrm{mix}(M)$ as a $*$-subalgebra of $\mathrm{LS}(M)$, particularly in relation to the absence of type II summands.
• To prove that all additive derivations on $\mathrm{mix}(M)$ are inner when $M$ is properly infinite, extending known results on inner derivations.

Proposed method

• Introduce $\mathrm{mix}(M)$ as a central extension of a von Neumann algebra $M$, showing it is a $C^*$-algebra over $S(Z(M)) \cong L^0(\Omega, \Sigma, \mu)$.
• Establish that $\mathrm{mix}(M) = \mathrm{LS}(M)$ if and only if $M$ has no type II direct summands.
• Use the topology of convergence locally in measure on $\mathrm{LS}(M)$ to analyze the behavior of derivations.
• Prove that additive derivations on $\mathrm{mix}(M)$ are $S(Z(M))$-linear, leveraging properties of the measure topology and spectral projections.
• Apply Sakai’s theorem to construct an element $a \in \mathrm{mix}(M)$ such that $D(x) = ax - xa$ for all $x \in \mathrm{mix}(M)$, proving innerness.
• Use decomposition via a sequence of orthogonal central projections $\{z_n\}$ with $\bigvee z_n = \mathbf{1}$ to localize the derivation and apply inner derivation structure on each $z_nM$.

Experimental results

Research questions

• RQ1 Under what conditions is every additive derivation on the algebra of locally measurable operators $\mathrm{LS}(M)$ inner?
• RQ2 How does the central extension $\mathrm{mix}(M)$ relate to $\mathrm{LS}(M)$, and when are they equal?
• RQ3 Can the innerness of additive derivations on $\mathrm{mix}(M)$ be established for properly infinite von Neumann algebras?
• RQ4 To what extent do the results on derivations in the noncommutative setting generalize previous results for commutative algebras like $L^0(0,1)$?
• RQ5 Is the $S(Z(M))$-linearity of derivations on $\mathrm{mix}(M)$ sufficient to ensure innerness in the properly infinite case?

Key findings

• The central extension $\mathrm{mix}(M)$ is a $*$-subalgebra of $\mathrm{LS}(M)$, and $\mathrm{mix}(M) = \mathrm{LS}(M)$ if and only if $M$ has no type II direct summands.
• Every additive derivation on $\mathrm{mix}(M)$ is $S(Z(M))$-linear, which is a key step toward proving innerness.
• For properly infinite von Neumann algebras, every additive derivation on $\mathrm{mix}(M)$ is inner, meaning it is implemented by an element in $\mathrm{mix}(M)$.
• This result implies that all additive derivations on $\mathrm{LS}(M)$ are inner when $M$ is of type I$_\infty$ or type III.
• The proof relies on decomposing $M$ into a direct sum of projections $\{z_n\}$ with $\bigvee z_n = \mathbf{1}$, allowing localization of the derivation to each $z_nM$.
• By applying Sakai’s theorem on each $z_nM$, an element $a \in \mathrm{mix}(M)$ is constructed such that $D(x) = ax - xa$ for all $x \in \mathrm{mix}(M)$, confirming innerness.