A local quantization principle for inclusions of tracial von Neumann algebras

2025/07/06 by Xinyan Cao, Junsheng Fang, Cao, Xinyan +5
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2507.04244

openalex publication_date 2025/07/06 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

We study the local quantization principle (after Sorin Popa~\citepopa 94 and \citepopa 95) of inclusions of tracial von Neumann algebras. Let (M,τ) be a type \rm II₁ von Neumann algebra and let N⊆ M be a type \rm II₁ von Neumann subalgebra. Let x₁,…, x_m ∈ M and ε> 0. Then there exists a partition of 1 with projections p₁, …, p_n in N such that ‖∑_i=1ⁿ p_i(x_j-E_{N'∩ M}(x_j))p_i‖₂lt;ε, 1≤ j≤ m. In particular, if N⊆ M is an inclusion of type \rm II₁ factors with [M:N]=2, then for any x₁,…, x_m∈ M, there exists a partition of 1 with projections p₁, …, p_n in N such that ∑_i=1ⁿ p_ix_jp_i=τ(x_j)1, 1≤ j≤ m. Equivalently, there exists a unitary operator u∈ N such that (1)/(n)∑_i=1ⁿu^*ix_j uⁱ=τ(x_j)1, 1≤ j≤ m.

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