On fundamental groups of tensor product \rm II₁ factors

2016/08/23 by Yusuke Isono, Isono, Yusuke · 8 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometry #Mathematics #Operator Algebras (math.OA) #Product (mathematics) #Pure mathematics #Spectral Theory in Mathematical Physics #Tensor product #math.OA

paper · pdf · doi:10.48550/arxiv.1608.06426

published in arXiv (Cornell University) (Cornell University) · 17 pages, final version, to appear in J. Inst. Math. Jussieu

openalex publication_date 2016/08/23 · arxiv created 2019/02/04 · arxiv updated 2019/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a \rm II₁ factor and let F(M) denote the fundamental group of M. In this article, we study the following property of M: for arbitrary \rm II₁ factor B, we have F(M ⊗ B)=F(M)F(B). We prove that for any subgroup G≤ ℝ^*₊ which is realized as a fundamental group of a \rm II₁ factor, there exists a \rm II₁ factor M which satisfies this property and whose fundamental group is G. Using this, we deduce that if G,H ≤ ℝ^*₊ are realized as fundamental groups of \rm II₁ factors (with separable predual), then so are groups G ⋅ H and G ∩ H.

Citations

• Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ � 1
• Explicit examples of equivalence relations and factors with prescribed fundamental group and outer automorphism group

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• On the Fundamental Group of Type II₁ Factors
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