A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911777676066816 |
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| author | Chandra, Harish Patel, Anurag Kumar |
| author_facet | Chandra, Harish Patel, Anurag Kumar |
| contents | We give a characterization of zero divisors of the ring $C[a,b].$ Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra $C[a,b].$ We also characterize the zero divisors and topological divisors of zero in $\ell^\infty.$ Further, we show that zero is the only zero divisor in the disk algebra $\mathscr{A}(\mathbb{D})$ and that the class of singular elements in $\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of $\mathscr{A}(\mathbb{D})$ which are not zero divisors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$ Chandra, Harish Patel, Anurag Kumar Functional Analysis Primary 13A70, 46H05 We give a characterization of zero divisors of the ring $C[a,b].$ Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra $C[a,b].$ We also characterize the zero divisors and topological divisors of zero in $\ell^\infty.$ Further, we show that zero is the only zero divisor in the disk algebra $\mathscr{A}(\mathbb{D})$ and that the class of singular elements in $\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of $\mathscr{A}(\mathbb{D})$ which are not zero divisors. |
| title | A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$ |
| topic | Functional Analysis Primary 13A70, 46H05 |
| url | https://arxiv.org/abs/2402.09909 |