On a problem of Johnson and Wolfe
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917540467310592 |
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| author | Maestre, Manuel García, Domingo Rodríguez-Vidanes, Daniel L. |
| author_facet | Maestre, Manuel García, Domingo Rodríguez-Vidanes, Daniel L. |
| contents | In 1979, Johnson and Wolfe proved that norm-attaining operators are dense in $L(C(K),C(S))$ when $K$ and $S$ are compact Hausdorff spaces in the real setting. The corresponding complex case has remained open since then, mainly because the real proof relies on order and sign-decomposition arguments that are no longer available for complex measures. In this paper, we settle the complex case. We prove that, for arbitrary compact Hausdorff spaces $K$ and $S$, the set of norm-attaining operators from the complex space $C(K)$ into the complex space $C(S)$ endowed with the supremum norm is dense in $L(C(K),C(S))$. The proof replaces the real order-theoretic mechanism by a measure-theoretic phase-correction argument, based on polar decompositions, unimodular approximation, and a semicontinuity principle for weighted total variation. This yields a complex defect-reduction procedure which recovers the Johnson-Wolfe density theorem in full generality for complex $C(K)$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a problem of Johnson and Wolfe Maestre, Manuel García, Domingo Rodríguez-Vidanes, Daniel L. Functional Analysis In 1979, Johnson and Wolfe proved that norm-attaining operators are dense in $L(C(K),C(S))$ when $K$ and $S$ are compact Hausdorff spaces in the real setting. The corresponding complex case has remained open since then, mainly because the real proof relies on order and sign-decomposition arguments that are no longer available for complex measures. In this paper, we settle the complex case. We prove that, for arbitrary compact Hausdorff spaces $K$ and $S$, the set of norm-attaining operators from the complex space $C(K)$ into the complex space $C(S)$ endowed with the supremum norm is dense in $L(C(K),C(S))$. The proof replaces the real order-theoretic mechanism by a measure-theoretic phase-correction argument, based on polar decompositions, unimodular approximation, and a semicontinuity principle for weighted total variation. This yields a complex defect-reduction procedure which recovers the Johnson-Wolfe density theorem in full generality for complex $C(K)$-spaces. |
| title | On a problem of Johnson and Wolfe |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2605.28466 |