A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras
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| Format: | Preprint |
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2024
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| _version_ | 1866909811208093696 |
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| author | Guo, Liang Qian, Jin Wang, Qin |
| author_facet | Guo, Liang Qian, Jin Wang, Qin |
| contents | Let $(X,d)$ be a uniformly locally finite metric space, and $T$ an operator in the uniform Roe algebra $C_u^*(X)$ (or uniform quasi-local algebra $C_{ql}^*(X)$). In this paper, we introduce the concept of limit operators of $T$ on galaxies in the nonstandard extension of $X$, and prove that $T$ is a generalized Fredholm operator with respect to the ghost ideal in $C_u^*(X)$ (or $C_{ql}^*(X)$) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if $X$ has Yu's Property A, then $T$ is a Fredholm operator if and only if all limit operators on afar galaxies are invertible. Using techniques in nonstandard analysis, our result strengthens a work of Špakula--Willett \cite{SpW} on the characterization of Fredholmness by using less limit operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras Guo, Liang Qian, Jin Wang, Qin Functional Analysis Metric Geometry Operator Algebras Let $(X,d)$ be a uniformly locally finite metric space, and $T$ an operator in the uniform Roe algebra $C_u^*(X)$ (or uniform quasi-local algebra $C_{ql}^*(X)$). In this paper, we introduce the concept of limit operators of $T$ on galaxies in the nonstandard extension of $X$, and prove that $T$ is a generalized Fredholm operator with respect to the ghost ideal in $C_u^*(X)$ (or $C_{ql}^*(X)$) if and only if all limit operators on afar galaxies are invertible, and their inverses are uniformly bounded. In particular, if $X$ has Yu's Property A, then $T$ is a Fredholm operator if and only if all limit operators on afar galaxies are invertible. Using techniques in nonstandard analysis, our result strengthens a work of Špakula--Willett \cite{SpW} on the characterization of Fredholmness by using less limit operators. |
| title | A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras |
| topic | Functional Analysis Metric Geometry Operator Algebras |
| url | https://arxiv.org/abs/2412.08130 |