The similarity problem and hyperreflexivity of von Neumann algebras
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| Format: | Preprint |
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2023
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| _version_ | 1866914642829246464 |
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| author | Eleftherakis, G. K. Papapetros, E. |
| author_facet | Eleftherakis, G. K. Papapetros, E. |
| contents | The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $\cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\colon \cl A\to \cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $\cl A$ satisfies the weak similarity property ((WSP) for short) if every $\mathrm{w}^*$-conitnuous unital and bounded homomorphism $u\colon \cl A\to \cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\cl A$ satisfies (WSP) if and only if the algebras $\cl A^{\prime}\bar \otimes \cl B(\ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $\cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\cl A^\prime \bar \otimes \cl B(\ell^2(\bb{N}))$ is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\cl A\bar \otimes \cl B,$ where $\cl A$ is an injective von Neumann algebra and $\cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $\text{w}^*$-similarity degree $d_{*}(\cl A\bar \otimes \cl B).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_07605 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The similarity problem and hyperreflexivity of von Neumann algebras Eleftherakis, G. K. Papapetros, E. Operator Algebras The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $\cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\colon \cl A\to \cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $\cl A$ satisfies the weak similarity property ((WSP) for short) if every $\mathrm{w}^*$-conitnuous unital and bounded homomorphism $u\colon \cl A\to \cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\cl A$ satisfies (WSP) if and only if the algebras $\cl A^{\prime}\bar \otimes \cl B(\ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $\cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\cl A^\prime \bar \otimes \cl B(\ell^2(\bb{N}))$ is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\cl A\bar \otimes \cl B,$ where $\cl A$ is an injective von Neumann algebra and $\cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $\text{w}^*$-similarity degree $d_{*}(\cl A\bar \otimes \cl B).$ |
| title | The similarity problem and hyperreflexivity of von Neumann algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2306.07605 |