Constructs the first examples of separable II₁ factors with no non-trivial crossed product decompositions via a novel embedding property into the tensor square.
Rigid Graph Products
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abstract
We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. Furthermore, we show that for many graph products of II$_1$-factors, including the hyperfinite II$_1$-factor, we can, up to a constant 2, retrieve the radius of the graph from the graph product. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products.
fields
math.OA 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit L_p-bounded Hilbert transforms, with the inequality equivalent to generation by finite-dimensional algebras of uniformly bounded dimension, extending free-product results and answering Ozawa's compactness probl
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A class of II$_1$ factors without non-trivial crossed product decompositions
Constructs the first examples of separable II₁ factors with no non-trivial crossed product decompositions via a novel embedding property into the tensor square.
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Hilbert transforms on graph products of finite von Neumann algebras
Graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit L_p-bounded Hilbert transforms, with the inequality equivalent to generation by finite-dimensional algebras of uniformly bounded dimension, extending free-product results and answering Ozawa's compactness probl