On spectral flow for operator algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914637169033216 |
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| author | Ng, Ping Wong Sutradhar, Arindam Wang, Cangyuan |
| author_facet | Ng, Ping Wong Sutradhar, Arindam Wang, Cangyuan |
| contents | Spectral flow was first studied by Atiyah and Lusztig, and first appeared in print in the work of Atiyah-Patodi-Singer (APS). For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra $\mathcal{M}(\mathcal{B})$ with $\mathcal{B}$ separable and stable, spectral flow roughly measures the ``net mass" of spectrum that passes through zero in the positive direction, as we move along the continuous path. As the index of a Fredholm operator has had many fruitful and important generalizations to general operator algebras, generalizing the spectral flow of a path of self-adjoint Fredholm operators would also be of great interest to operator theory. We develop a notion of spectral flow which works for arbitrary separable stable canonical ideals -- including stably projectionless C*-algebras (which depends on a quite general notion of essential codimension). We show that, under appropriate hypotheses, spectral flow induces a group isomorphism $π_1(Fred_{SA,\infty},pt)\cong K_0(\mathcal{B})$, generalizing a result of APS. We also provide an axiomatization of spectral flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12061 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On spectral flow for operator algebras Ng, Ping Wong Sutradhar, Arindam Wang, Cangyuan Operator Algebras Functional Analysis 46L35, 46L80, 46L87, 47A53 Spectral flow was first studied by Atiyah and Lusztig, and first appeared in print in the work of Atiyah-Patodi-Singer (APS). For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra $\mathcal{M}(\mathcal{B})$ with $\mathcal{B}$ separable and stable, spectral flow roughly measures the ``net mass" of spectrum that passes through zero in the positive direction, as we move along the continuous path. As the index of a Fredholm operator has had many fruitful and important generalizations to general operator algebras, generalizing the spectral flow of a path of self-adjoint Fredholm operators would also be of great interest to operator theory. We develop a notion of spectral flow which works for arbitrary separable stable canonical ideals -- including stably projectionless C*-algebras (which depends on a quite general notion of essential codimension). We show that, under appropriate hypotheses, spectral flow induces a group isomorphism $π_1(Fred_{SA,\infty},pt)\cong K_0(\mathcal{B})$, generalizing a result of APS. We also provide an axiomatization of spectral flow. |
| title | On spectral flow for operator algebras |
| topic | Operator Algebras Functional Analysis 46L35, 46L80, 46L87, 47A53 |
| url | https://arxiv.org/abs/2312.12061 |