On closed Lie ideals of certain tensor products of $C^*$-algebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866912795428126720 |
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| author | Jain, Ranjana Gupta, Ved Prakash |
| author_facet | Jain, Ranjana Gupta, Ved Prakash |
| contents | For a simple $C^*$-algebra $A$ and any other $C^*$-algebra $B$, it is proved that every closed ideal of $A \otimes^{\min} B$ is a product ideal if either $A$ is exact or $B$ is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi-central approximate identity is described in terms of the commutator of the Banach algebra. If $α$ is either the Haagerup norm, the operator space projective norm or the $C^*$-minimal norm, then this allows us to identify all closed Lie ideals of $A \otimes^α B$, where $A$ and $B$ are simple, unital $C^*$-algebras with one of them admitting no tracial functionals, and to deduce that every non-central closed Lie ideal of $B(H) \otimes^α B(H)$ contains the product ideal $K(H) \otimes^α K(H)$. Closed Lie ideals of $A \otimes^{\min} C(X)$ are also determined, $A$ being any simple unital $C^*$-algebra with at most one tracial state and $X$ any compact Hausdorff space. And, it is shown that closed Lie ideals of $A \otimes^α K(H)$ are precisely the product ideals, where $A$ is any unital $C^*$-algebra and $α$ any completely positive uniform tensor norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1701_02218 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On closed Lie ideals of certain tensor products of $C^*$-algebras Jain, Ranjana Gupta, Ved Prakash Operator Algebras 46L06 For a simple $C^*$-algebra $A$ and any other $C^*$-algebra $B$, it is proved that every closed ideal of $A \otimes^{\min} B$ is a product ideal if either $A$ is exact or $B$ is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi-central approximate identity is described in terms of the commutator of the Banach algebra. If $α$ is either the Haagerup norm, the operator space projective norm or the $C^*$-minimal norm, then this allows us to identify all closed Lie ideals of $A \otimes^α B$, where $A$ and $B$ are simple, unital $C^*$-algebras with one of them admitting no tracial functionals, and to deduce that every non-central closed Lie ideal of $B(H) \otimes^α B(H)$ contains the product ideal $K(H) \otimes^α K(H)$. Closed Lie ideals of $A \otimes^{\min} C(X)$ are also determined, $A$ being any simple unital $C^*$-algebra with at most one tracial state and $X$ any compact Hausdorff space. And, it is shown that closed Lie ideals of $A \otimes^α K(H)$ are precisely the product ideals, where $A$ is any unital $C^*$-algebra and $α$ any completely positive uniform tensor norm. |
| title | On closed Lie ideals of certain tensor products of $C^*$-algebras |
| topic | Operator Algebras 46L06 |
| url | https://arxiv.org/abs/1701.02218 |