[Paper Review] A class of superrigid group von Neumann algebras

Adrian Ioana, Sorin Popa+1 more|arXiv (Cornell University)|Jul 8, 2010

Advanced Operator Algebra Research32 references4 citations

TL;DR

This paper establishes the first W*-superrigidity result for group von Neumann algebras by proving that for a broad class of generalized wreath product groups, the von Neumann algebra LG completely remembers the group G: if LG is isomorphic to LΛ for another countable group Λ, then G and Λ must be isomorphic. The proof relies on deformation/rigidity techniques, including asymptotic orthogonality and the use of cohomological invariants in crossed product structures.

ABSTRACT

We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra L(G) completely "remembers" the group G. More precisely, if L(G) is isomorphic to the von Neumann algebra L(Λ) of an arbitrary countable group Λ, then Λ must be isomorphic to G. This represents the first superrigidity result pertaining to group von Neumann algebras.

Motivation & Objective

• To establish W*-superrigidity for a large class of group von Neumann algebras, proving that LG uniquely determines the group G up to isomorphism.
• To extend the scope of superrigidity beyond C*-algebras and II₁ factors by focusing on the von Neumann algebraic structure of group algebras.
• To address Connes' long-standing conjecture on the uniqueness of group factors for icc property (T) groups by proving it for a broad class of wreath product groups.
• To develop and apply new techniques in deformation/rigidity theory to analyze the structure of crossed products and asymptotic orthogonality in von Neumann algebras.
• To show that the group structure is encoded in the weak closure of the group algebra, even when the group is non-amenable or has property (T).

Proposed method

• Use of the comultiplication Δ on LG and its conjugation via a unitary v to transfer the problem into a crossed product structure M ⊗ M = (M_d(ℂ) ⊗ M ⊗ A) ⋊ Γ.
• Application of the Fourier decomposition with respect to the group Γ to analyze the asymptotic behavior of matrix coefficients in the crossed product algebra.
• Employment of the relative position relation x ≺ B to detect non-amenable subalgebras and to rule out the existence of certain embeddings via asymptotic orthogonality.
• Construction of a normal *-homomorphism θ: A → (M_d(ℂ) ⊗ A ⊗ A)^AdK to relate the comultiplication Δ to the group action and to analyze the structure of the image under Δ.
• Use of the unitary v ∈ M ⊗ M to conjugate the comultiplication Δ so that Δ(b) = v(b ⊗ 1)v* for b ∈ B, enabling control over the Fourier coefficients.
• Proof by contradiction: assuming B ≺ A leads to the conclusion that the projection p must be zero, contradicting the initial assumption that p ≠ 0, thus establishing the non-embeddability of B into A.

Experimental results

Research questions

• RQ1 Can the group von Neumann algebra LG uniquely determine the group G up to isomorphism for a broad class of non-amenable groups?
• RQ2 Does W*-superrigidity hold for generalized wreath product groups, particularly those with property (T) or infinite conjugacy classes?
• RQ3 To what extent does the weak closure of the group algebra retain structural information about the original group, especially when the group is non-abelian and non-amenable?
• RQ4 Can deformation/rigidity techniques be used to prove that isomorphisms between group von Neumann algebras imply isomorphisms between the underlying groups?
• RQ5 Is it possible to rule out the existence of non-trivial embeddings of subalgebras into abelian subalgebras using asymptotic orthogonality of Fourier coefficients?

Key findings

• For any group G in a large class of generalized wreath product groups, LG ≅ LΛ implies G ≅ Λ, establishing the first W*-superrigidity result for group von Neumann algebras.
• The proof relies on the deformation/rigidity framework, particularly the use of asymptotic orthogonality of Fourier coefficients in crossed product von Neumann algebras.
• The authors show that if B ≺ A, then the Fourier coefficients of (b_n ⊗ 1)vp tend to zero, leading to a contradiction unless p = 0, thus proving the non-embeddability of B into A.
• The structure of the comultiplication Δ and its conjugation via a unitary v allows the authors to transfer the problem into a setting where relative position and asymptotic orthogonality can be analyzed.
• The result confirms Connes' conjecture for this class of groups: LΛ ≅ LG with G having property (T) implies Λ ≅ G.
• The method successfully distinguishes group factors arising from non-isomorphic groups, even when the groups are non-amenable or have infinite conjugacy classes.