A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913311250972672 |
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| author | Ding, Changying |
| author_facet | Ding, Changying |
| contents | We show that if $Γ\curvearrowright (X^Γ,μ^Γ)$ is a Bernoulli action of an i.c.c. nonamenable group $Γ$ which is weakly amenable with Cowling-Haagerup constant $1$, and $Λ\curvearrowright(Y,ν)$ is a free ergodic p.m.p. algebraic action of a group $Λ$, then the isomorphism $L^\infty(X^Γ)\rtimesΓ\cong L^\infty(Y)\rtimesΛ$ implies that $L^\infty(X^Γ)$ and $L^\infty(Y)$ are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from \cite{BIP21}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups Ding, Changying Operator Algebras Group Theory We show that if $Γ\curvearrowright (X^Γ,μ^Γ)$ is a Bernoulli action of an i.c.c. nonamenable group $Γ$ which is weakly amenable with Cowling-Haagerup constant $1$, and $Λ\curvearrowright(Y,ν)$ is a free ergodic p.m.p. algebraic action of a group $Λ$, then the isomorphism $L^\infty(X^Γ)\rtimesΓ\cong L^\infty(Y)\rtimesΛ$ implies that $L^\infty(X^Γ)$ and $L^\infty(Y)$ are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from \cite{BIP21}. |
| title | A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups |
| topic | Operator Algebras Group Theory |
| url | https://arxiv.org/abs/2404.08182 |