Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2501.07544 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910985022865408 |
|---|---|
| author | Clouâtre, Raphaël Krisko, Colin |
| author_facet | Clouâtre, Raphaël Krisko, Colin |
| contents | Let $S$ be a concrete operator system represented on some Hilbert space $H$. A $C^*$-support of $S$ is the $C^*$-algebra generated (via the Choi--Effros product) by $S$ inside an injective operator system acting on $H$. By leveraging Hamana's theory, we show that such a $C^*$-support is unique precisely when $C^*(S)$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^*$-supports can be used to give new characterizations of the unique extension property for $*$-representations, as well as the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^*$-supports of $S$ to describe the subspace generated by the so-called abnormalities of $S$, thereby complementing a result of Kakariadis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $C^*$-supports and abnormalities of operator systems Clouâtre, Raphaël Krisko, Colin Operator Algebras Functional Analysis Let $S$ be a concrete operator system represented on some Hilbert space $H$. A $C^*$-support of $S$ is the $C^*$-algebra generated (via the Choi--Effros product) by $S$ inside an injective operator system acting on $H$. By leveraging Hamana's theory, we show that such a $C^*$-support is unique precisely when $C^*(S)$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^*$-supports can be used to give new characterizations of the unique extension property for $*$-representations, as well as the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^*$-supports of $S$ to describe the subspace generated by the so-called abnormalities of $S$, thereby complementing a result of Kakariadis. |
| title | $C^*$-supports and abnormalities of operator systems |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2501.07544 |