[Paper Review] Lie derivations on the algebras of locally measurable operators

Vladimir Chilin, I. T. Juraev|arXiv (Cornell University)|Aug 13, 2016

Advanced Topics in Algebra3 references3 citations

TL;DR

This paper establishes that every Lie derivation on a solid *-subalgebra of locally measurable operators affiliated with a von Neumann algebra decomposes uniquely into the sum of an inner derivation, a derivation induced by a center-valued derivation, and a center-valued trace. The result extends the standard form of Lie derivations from C*-algebras and von Neumann algebras to the broader class of algebras of locally measurable operators, providing a complete structural characterization in the type I case.

ABSTRACT

We prove that every Lie derivation on a solid $\star-$subalgebras of locally measurable operators it is equal to a sum of the associative derivation and the center-valued trace.

Motivation & Objective

• To characterize the structure of Lie derivations on solid *-subalgebras of locally measurable operators affiliated with von Neumann algebras.
• To extend the standard form decomposition of Lie derivations—known for C*-algebras and von Neumann algebras—to the more general setting of locally measurable operator algebras.
• To establish that every Lie derivation on such algebras admits a unique representation as the sum of an inner derivation, a derivation induced by a center-valued derivation, and a center-valued trace.
• To provide a complete structural classification of Lie derivations in the case of type I von Neumann algebras, including finite and properly infinite factors.

Proposed method

• The authors use the theory of locally measurable operators and their algebras LS(M), which generalize the algebras S(M) of measurable operators.
• They define solid *-subalgebras of LS(M) as those closed under multiplication by elements of the von Neumann algebra M from both sides.
• The key technique involves decomposing derivations on LS(M) using central projections and reducing the problem to finite-type I von Neumann algebras via direct sum decompositions.
• For finite type I algebras, derivations are constructed via matrix algebras over the center’s measurable operators, using derivations on the center’s algebra S(Z).
• The derivation Dδ is explicitly defined on the product algebra ∏_{n∈F} LS(z_nM) by applying the center-derivation δ_n to each matrix entry via formula (4.1).
• The final derivation D is extended to the full LS(M) by setting Dδ(x₁ + x₂) = Dδ(x₁) for x₁ ∈ z₀LS(M) and x₂ ∈ z₀⊥LS(M), leveraging known results on inner derivations in the properly infinite part.

Experimental results

Research questions

• RQ1 Can the standard form of Lie derivations—known for C*-algebras and von Neumann algebras—be extended to algebras of locally measurable operators?
• RQ2 What is the precise structural decomposition of a Lie derivation on a solid *-subalgebra of LS(M) containing M?
• RQ3 How do derivations on the center of LS(M) induce derivations on the entire algebra, and what role do they play in the decomposition?
• RQ4 To what extent does the decomposition of Lie derivations depend on the type I structure of the underlying von Neumann algebra?
• RQ5 Is the decomposition of a Lie derivation into inner derivation, center-derivation, and center-valued trace unique in the context of LS(M)?

Key findings

• Every Lie derivation on a solid *-subalgebra of LS(M) containing M admits a unique decomposition into the sum of an inner derivation, a derivation induced by a center-valued derivation, and a center-valued trace.
• In the case of a type I von Neumann algebra M, the decomposition takes the form L = D_a + D_δ + E, where D_a is inner, D_δ arises from a derivation δ on the center S(Z), and E is a center-valued trace.
• The derivation D_δ is explicitly constructed via matrix entries using the formula D_δ(∑λ_ij e_ij) = ∑δ(λ_ij)e_ij, where δ acts on the center S(Z).
• For finite type I von Neumann algebras, LS(M) is isomorphic to a product of matrix algebras over L^0-spaces, enabling the construction of uncountably many distinct derivations when the center has no atoms.
• The restriction of any derivation D on LS(M) to the center S(Z) determines the entire D_δ component, and D_δ is trivial on the properly infinite part when restricted via central projections.
• The result confirms that the standard form of Lie derivations holds in LS(M), generalizing known results from S(M) and C*-algebras to the non-bounded setting of locally measurable operators.