$K$-stability of $C^*$-algebras generated by isometries and unitaries with twisted commutation relations
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arXiv
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| Format: | Preprint |
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2023
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| author | Bhatt, Shreema Subhash Saurabh, Bipul |
| author_facet | Bhatt, Shreema Subhash Saurabh, Bipul |
| contents | In this article, we prove $K$-stability for a family of $C^*$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^*$-algebra of doubly non-commuting isometries and free twist of isometries.
Next, we consider the $C^*$-algebra $A_{\mathcal{V}}$ generated by an $n$-tuple of $\mathcal{U}$-twisted isometries $\mathcal{V}$ with respect to a fixed $n\choose 2$-tuple $\mathcal{U}=\{U_{ij}:1\leq i<j \leq n\}$ of commuting unitaries (see \cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^*$-algebra generated by $(\{U_{ij}:1\leq i<j \leq n\})$ does not contain any element of finite order in the torus group $\bbbt^{n\choose 2}$, we show that $A_{\mathcal{V}}$
is $K$-stable. Finally, we prove the same result for the $C^*$-algebra generated by a tuple of free $\mathcal{U}$-twisted isometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06189 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $K$-stability of $C^*$-algebras generated by isometries and unitaries with twisted commutation relations Bhatt, Shreema Subhash Saurabh, Bipul Operator Algebras K-Theory and Homology 58B32, 58B34, 46L89, 47L55 In this article, we prove $K$-stability for a family of $C^*$-algebras, which are generated by a finite set of unitaries and isometries satisfying twisted commutation relations. This family includes the $C^*$-algebra of doubly non-commuting isometries and free twist of isometries. Next, we consider the $C^*$-algebra $A_{\mathcal{V}}$ generated by an $n$-tuple of $\mathcal{U}$-twisted isometries $\mathcal{V}$ with respect to a fixed $n\choose 2$-tuple $\mathcal{U}=\{U_{ij}:1\leq i<j \leq n\}$ of commuting unitaries (see \cite{NarJaySur-2022aa}). Under the assumption that the spectrum of the commutative $C^*$-algebra generated by $(\{U_{ij}:1\leq i<j \leq n\})$ does not contain any element of finite order in the torus group $\bbbt^{n\choose 2}$, we show that $A_{\mathcal{V}}$ is $K$-stable. Finally, we prove the same result for the $C^*$-algebra generated by a tuple of free $\mathcal{U}$-twisted isometries. |
| title | $K$-stability of $C^*$-algebras generated by isometries and unitaries with twisted commutation relations |
| topic | Operator Algebras K-Theory and Homology 58B32, 58B34, 46L89, 47L55 |
| url | https://arxiv.org/abs/2312.06189 |