W*-representations of subfactors and restrictions on the Jones index

2021/12/30 by Sorin Popa, Popa, Sorin
Materials Science · Mathematics · #46L36 #46L37 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Organic and Molecular Conductors Research

paper · pdf · doi:10.48550/arxiv.2112.15148

openalex publication_date 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A \it W^*-representation of a II₁ subfactor N⊂ M with finite Jones index, [M:N]<∞, is a non-degenerate commuting square embedding of N⊂ M into an inclusion of atomic von Neumann algebras ⊕_{i∈ I} \Cal B(\Cal K_i)=\Cal N ⊂^{\Cal E} \Cal M=⊕_{j∈ J} \Cal B(\Cal H_j). We undertake here a systematic study of this notion, first introduced in [P92], giving examples and considering invariants such as the (bipartite) \it inclusion graph Λ_{\Cal N ⊂ \Cal M}, the \it coupling vector (\rm dim(_M\Cal H_j))_j and the \it RC-algebra (relative commutant) M'∩ \Cal N, for which we establish some basic properties. We then prove that if N⊂ M admits a W^*-representation \Cal N⊂^{\Cal E}\Cal M, with the expectation \Cal E preserving a semifinite trace on \Cal M, such that there exists a norm one projection of \Cal M onto M commuting with \Cal E, a property of N⊂ M that we call \it weak injectivity/amenability, then [M:N] equals the square norm of the inclusion graph Λ_{\Cal N ⊂ \Cal M}.

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