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Internal graphs of graph products of hyperfinite II$_1$-factors

Martijn Caspers, Enli Chen

TL;DR

The paper proves that for graphs Γ in the $H$-rigid class, the internal graph $Int(Γ)$ is determined by the graph product $R_Γ$ of hyperfinite II$_1$-factors, yielding rigidity results and enabling classification for specific graphs like lines and cycles. The approach relies on the Peterson–Thom conjecture (now established) and Popa’s intertwining-by-bimodule theory to transfer embeddings from vertex algebras to subgraphs, ultimately showing $Int(Γ)\cong Int(Λ)$ whenever $R_Γ\cong R_Λ$ under suitable hypotheses. This leads to concrete classifications (e.g., $R_{l_n}$, $R_{\mathbb{Z}_n}$) and strengthens radius rigidity, showing the radius difference of isomorphic graph products is at most $1$ under the $H$-rigid framework. The results advance rigidity theory for amenable vertex factors and highlight the interior graph as a robust invariant in graph product von Neumann algebras.

Abstract

In this paper, we show that for a graph $Γ$ from a class named H-rigid graphs, its subgraph ${\rm Int}(Γ)$, named the internal graph of $Γ$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_Γ$. In particular, we can classify $R_Γ$ for some typical types of graphs, such as lines and cyclic graphs. As an application, we also show that for two isomorphic graph products of hyperfinite II$_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.

Internal graphs of graph products of hyperfinite II$_1$-factors

TL;DR

The paper proves that for graphs Γ in the -rigid class, the internal graph is determined by the graph product of hyperfinite II-factors, yielding rigidity results and enabling classification for specific graphs like lines and cycles. The approach relies on the Peterson–Thom conjecture (now established) and Popa’s intertwining-by-bimodule theory to transfer embeddings from vertex algebras to subgraphs, ultimately showing whenever under suitable hypotheses. This leads to concrete classifications (e.g., , ) and strengthens radius rigidity, showing the radius difference of isomorphic graph products is at most under the -rigid framework. The results advance rigidity theory for amenable vertex factors and highlight the interior graph as a robust invariant in graph product von Neumann algebras.

Abstract

In this paper, we show that for a graph from a class named H-rigid graphs, its subgraph , named the internal graph of , is an isomorphism invariant of the graph product of hyperfinite II-factors . In particular, we can classify for some typical types of graphs, such as lines and cyclic graphs. As an application, we also show that for two isomorphic graph products of hyperfinite II-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.
Paper Structure (12 sections, 13 theorems, 22 equations, 1 figure)

This paper contains 12 sections, 13 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normalizers, strong solidity and quasi-strong solidity
  4. Popa's intertwining-by-bimodule theory
  5. Simple graphs
  6. Particular graphs
  7. Radius of graphs
  8. Graph products
  9. The Peterson-Thom conjecture and quasi-strong solidity of $R_{\Gamma}$
  10. Main theorem
  11. H-rigid graphs
  12. Graph products of hyperfinite II$_1$-factors over H-rigid graphs

Key Result

Lemma 2.1

Suppose $M$ and $N$ are two von Neumann algebras, and $A\subseteq M$ and $B\subseteq N$ are von Neumann subalgebras. Let $x\in M\overline{\otimes} N$. If for any $\omega\in N_*$, $(id\otimes \omega)(x)\in A$, and for any $\omega'\in M_*$, $(\omega'\otimes id)(x)\in B$, then $x\in A\overline{\otimes}

Figures (1)

  • Figure 1: The left is $l_5$, the right is $\mathbb{Z}_5$

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Definition 2.2: Strong solidity and quasi-strong solidity
  • Definition 2.3: Embedding $A\prec_M B$
  • Definition 2.4: Stable embedding $A\prec^s_M B$
  • Lemma 2.5: Lemma 2.4 in drimbePrimeII1Factors2019, see also vaesExplicitComputationsAll2008
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 3.1: Theorem 1.3 of HayesJekelElayavalli
  • ...and 24 more