[Paper Review] Local And 2-Local Derivations On Algebras Of Measurable Operators

Shavkat Ayupov, Karimbergen Kudaybergenov|arXiv (Cornell University)|Feb 16, 2016

Advanced Operator Algebra Research34 references3 citations

TL;DR

This paper investigates derivations, local derivations, and 2-local derivations on algebras of measurable operators affiliated with von Neumann algebras. It establishes that every 2-local derivation on finite type I von Neumann algebras without abelian summands and on type I∞ algebras is automatically a derivation, while providing examples in commutative settings where 2-local derivations fail to be derivations.

ABSTRACT

The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II$_1$. In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra $M$, under which every local or 2-local derivation on the algebra of measurable operators affiliated with $M$ is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations.

Motivation & Objective

• To characterize derivations, local derivations, and 2-local derivations on algebras of measurable operators affiliated with von Neumann algebras.
• To determine conditions under which 2-local derivations on such algebras are necessarily derivations.
• To identify cases where 2-local derivations are not derivations, particularly in commutative settings.
• To extend results on 2-local derivations from bounded operators to unbounded measurable operator algebras.
• To examine the role of measure continuity and type classification (e.g., type II₁) in the structure of derivations.

Proposed method

• Analyzes derivations on algebras of measurable operators affiliated with von Neumann algebras, distinguishing cases by type (I, II₁, I∞).
• Uses the concept of locally measurable operators and the extended center-valued trace Φ for type I∞ algebras.
• Applies the identity Φ(Δ(x)y) = -Φ(xΔ(y)) for finite-range operators in LS(M) to prove that 2-local derivations are derivations.
• Employs structural lemmas to show that 2-local derivations on matrix algebras over commutative regular algebras are derivations.
• Constructs counterexamples in abelian von Neumann algebras where the projection lattice is not atomic to show non-derivation 2-local derivations.
• Reduces the problem to studying derivations on the center and matrix entries via decomposition into matrix units e_ij.

Experimental results

Research questions

• RQ1 Under what conditions on a von Neumann algebra M is every 2-local derivation on S(M) necessarily a derivation?
• RQ2 Can 2-local derivations on commutative algebras of measurable operators fail to be derivations, and if so, under what conditions?
• RQ3 How does the type of the von Neumann algebra (e.g., type I, I∞, II₁) affect the structure of derivations and 2-local derivations?
• RQ4 What role does measure continuity play in the automatic derivativeness of derivations on type II₁ algebras?
• RQ5 Is every 2-local derivation on LS(M) a derivation when M is a finite type I von Neumann algebra without abelian summands?

Key findings

• Every 2-local derivation on the algebra LS(M) = S(M) of measurable operators affiliated with a finite von Neumann algebra M of type I without abelian direct summands is a derivation.
• Every 2-local derivation on the algebra LS(M) of a type I∞ von Neumann algebra M is a derivation, as shown via the extended center-valued trace and key trace identity.
• For matrix algebras M_n(𝒜) over a commutative regular algebra 𝒜 with n ≥ 2, every 2-local derivation is a derivation, due to structural lemmas on matrix units and center restrictions.
• In the abelian case, if the lattice of projections P(M) is not atomic, then S(M) admits a 2-local derivation that is not a derivation.
• A 2-local derivation Δ on a *-subalgebra 𝒞 of LS(M) containing M is a derivation if M is of type I∞, as established by trace-based identities.
• The paper provides a complete description of derivations on algebras of measurable operators for all von Neumann algebra types except type II₁, where measure continuity is required.