Morita theory for dynamical von Neumann algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913932935954432 |
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| author | De Ro, Joeri |
| author_facet | De Ro, Joeri |
| contents | Given a locally compact quantum group $\mathbb{G}$ and two $\mathbb{G}$-$W^*$-algebras $α: A\curvearrowleft \mathbb{G}$ and $β: B\curvearrowleft \mathbb{G}$, we study the notion of equivariant $W^*$-Morita equivalence $(A, α)\sim_{\mathbb{G}} (B, β)$, which is an equivariant version of Rieffel's notion of $W^*$-Morita equivalence. We prove that important dynamical properties of $\mathbb{G}$-$W^*$-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra $L^\infty(\mathbb{K}\backslash \mathbb{G})\subseteq L^\infty(\mathbb{G})$ with dual coideal von Neumann algebra $L^\infty(\check{\mathbb{K}})\subseteq L^\infty(\check{\mathbb{G}})$, we use a natural $\check{\mathbb{G}}$-$W^*$-Morita equivalence $L^\infty(\mathbb{K}\backslash \mathbb{G})\rtimes_Δ\mathbb{G} \sim_{\check{\mathbb{G}}} L^\infty(\check{\mathbb{K}})$ to relate dynamical properties of $L^\infty(\mathbb{K}\backslash \mathbb{G})$ with dynamical properties of $L^\infty(\check{\mathbb{K}})$. We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski and Vergnioux, namely that for $\mathbb{H}$ a closed quantum subgroup of the compact quantum group $\mathbb{G}$, coamenability of $\mathbb{H}\backslash \mathbb{G}$ and relative amenability of $\ell^\infty(\check{\mathbb{H}})$ in $\ell^\infty(\check{\mathbb{G}})$ are equivalent. Moreover, if $\mathbb{G}$ is compact, we study the relation between $\mathbb{G}$-$W^*$-Morita equivalence of $(A, α)$ and $(B, β)$ and $\mathbb{G}$-$C^*$-Morita equivalence of the associated $\mathbb{G}$-$C^*$-algebras $(\mathcal{R}(A), α)$ and $(\mathcal{R}(B), β)$ of regular elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17407 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morita theory for dynamical von Neumann algebras De Ro, Joeri Operator Algebras Quantum Algebra Given a locally compact quantum group $\mathbb{G}$ and two $\mathbb{G}$-$W^*$-algebras $α: A\curvearrowleft \mathbb{G}$ and $β: B\curvearrowleft \mathbb{G}$, we study the notion of equivariant $W^*$-Morita equivalence $(A, α)\sim_{\mathbb{G}} (B, β)$, which is an equivariant version of Rieffel's notion of $W^*$-Morita equivalence. We prove that important dynamical properties of $\mathbb{G}$-$W^*$-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra $L^\infty(\mathbb{K}\backslash \mathbb{G})\subseteq L^\infty(\mathbb{G})$ with dual coideal von Neumann algebra $L^\infty(\check{\mathbb{K}})\subseteq L^\infty(\check{\mathbb{G}})$, we use a natural $\check{\mathbb{G}}$-$W^*$-Morita equivalence $L^\infty(\mathbb{K}\backslash \mathbb{G})\rtimes_Δ\mathbb{G} \sim_{\check{\mathbb{G}}} L^\infty(\check{\mathbb{K}})$ to relate dynamical properties of $L^\infty(\mathbb{K}\backslash \mathbb{G})$ with dynamical properties of $L^\infty(\check{\mathbb{K}})$. We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski and Vergnioux, namely that for $\mathbb{H}$ a closed quantum subgroup of the compact quantum group $\mathbb{G}$, coamenability of $\mathbb{H}\backslash \mathbb{G}$ and relative amenability of $\ell^\infty(\check{\mathbb{H}})$ in $\ell^\infty(\check{\mathbb{G}})$ are equivalent. Moreover, if $\mathbb{G}$ is compact, we study the relation between $\mathbb{G}$-$W^*$-Morita equivalence of $(A, α)$ and $(B, β)$ and $\mathbb{G}$-$C^*$-Morita equivalence of the associated $\mathbb{G}$-$C^*$-algebras $(\mathcal{R}(A), α)$ and $(\mathcal{R}(B), β)$ of regular elements. |
| title | Morita theory for dynamical von Neumann algebras |
| topic | Operator Algebras Quantum Algebra |
| url | https://arxiv.org/abs/2410.17407 |