Takesaki duality for weak* closed $L^p$-operator crossed products
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917417946447872 |
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| author | Wang, Zhen |
| author_facet | Wang, Zhen |
| contents | The aim of this paper is to study Takesaki duality for weak* closed $L^p$-operator crossed product $W^*_p(G,A,α)$, where $G$ is a countable discrete Abelian group, $A$ is a unital separable weak* closed $L^p$-operator algebra ($p>1$), and $α$ is a weak* continuous $p$-completely isometric action of $G$ on $A$. In this paper, we construct a weak* continuous homomorphism $Φ$ from $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$. We show that $Φ$ is an isomorphism if and only if either $p=2$ or $G$ is finite, and $Φ$ is an isometric isomorphism if either $p=2$ or $G$ is trivial. It is also proved that $Φ$ is equivariant for the double dual action $\hat{\hatα}$ of $G$ on $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ and the action $\mathrm{Ad}ρ_p\otimesα$ of $G$ on $\mathcal{B}(l^p(G))\bar{\otimes} A$. Furthermore, we prove that $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ is weak* continuous isometrically isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is trivial, and $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ is weak* continuous isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is finite when $A=M_n^p$. This shows that Takesaki duality theorem of von Neumann algebras can be generalized to weak* closed $L^2$-operator algebras, and this theorem can not be generalized to weak* closed $L^p$-operator algebras when $p\in (1,\infty)\setminus\{2\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16946 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Takesaki duality for weak* closed $L^p$-operator crossed products Wang, Zhen Functional Analysis Operator Algebras The aim of this paper is to study Takesaki duality for weak* closed $L^p$-operator crossed product $W^*_p(G,A,α)$, where $G$ is a countable discrete Abelian group, $A$ is a unital separable weak* closed $L^p$-operator algebra ($p>1$), and $α$ is a weak* continuous $p$-completely isometric action of $G$ on $A$. In this paper, we construct a weak* continuous homomorphism $Φ$ from $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$. We show that $Φ$ is an isomorphism if and only if either $p=2$ or $G$ is finite, and $Φ$ is an isometric isomorphism if either $p=2$ or $G$ is trivial. It is also proved that $Φ$ is equivariant for the double dual action $\hat{\hatα}$ of $G$ on $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ and the action $\mathrm{Ad}ρ_p\otimesα$ of $G$ on $\mathcal{B}(l^p(G))\bar{\otimes} A$. Furthermore, we prove that $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ is weak* continuous isometrically isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is trivial, and $W^*_p(\hat{G},W^*_p(G,A,α),\hatα)$ is weak* continuous isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is finite when $A=M_n^p$. This shows that Takesaki duality theorem of von Neumann algebras can be generalized to weak* closed $L^2$-operator algebras, and this theorem can not be generalized to weak* closed $L^p$-operator algebras when $p\in (1,\infty)\setminus\{2\}$. |
| title | Takesaki duality for weak* closed $L^p$-operator crossed products |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2604.16946 |