The von Neumann algebra characterization theorems (Q1065327)
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scientific article; zbMATH DE number 3921283
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The von Neumann algebra characterization theorems |
scientific article; zbMATH DE number 3921283 |
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The von Neumann algebra characterization theorems (English)
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1985
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There are two standard characterization theorems that give necessary and sufficient conditions for a \(C^*\)-algebra to be isomorphic to a von Neumann algebra. Sakai's states that these \(C^*\)-algebras are precisely those that are (isometric to) the norm-dual space of some Banach space. The author's states that these \(C^*\)-algebras are the ones that are monotone closed and possess a separating family of normal states. In this paper, the author derives the first characterization from the second. Complete, streamlined proofs are given.
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necessary and sufficient conditions for a \(C^*\)-algebra to be
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isomorphic to a von Neumann algebra
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monotone closed
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separating family of normal states
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necessary and sufficient conditions for a \(C^*\)- algebra to be isomorphic to a von Neumann algebra
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