On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917358929444864 |
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| author | Karak, Sainik |
| author_facet | Karak, Sainik |
| contents | This article aims to examine the Hahn-Banach smoothness of Banach spaces and its connections to various geometrical aspects. We examine the circumstances that allow linear functionals to have unique norm-preserving extensions, with particular attention to the behavior of these properties in $L_1$-preduals and in spaces of affine continuous functions. Banach spaces which are $L_1$-preduals and also Hahn-Banach smooth are completely characterized. It is demonstrated that if $X$ is an $M$-embedded space then $X^*$ admits a predual which is not weakly Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ and also represents a split face, no subspace of $A(S)$ retains the property-$(wU)$ in $A(S)^{**}$. Furthermore, when $X=C_0(L)$, in the context of a locally compact Hausdorff space $L$, the continuity of the identity mapping $I:(B_{X^*},w^*)\to (B_{X^*},w)$ in $ext (B_{X^*})$ significantly influences the subspaces of $X$ that have unique extension property in $X^{**}$. Collectively, this study provides structural characterizations of specialized geometric property, so called Hahn-Banach smoothness, and offers solutions to some natural problems enlisted at the beginning that involve spaces that are $L_1$-preduals and also spaces that are $M$-embedded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces Karak, Sainik Functional Analysis 46A22, 46B20, 46E15, 46B22, 46M05 This article aims to examine the Hahn-Banach smoothness of Banach spaces and its connections to various geometrical aspects. We examine the circumstances that allow linear functionals to have unique norm-preserving extensions, with particular attention to the behavior of these properties in $L_1$-preduals and in spaces of affine continuous functions. Banach spaces which are $L_1$-preduals and also Hahn-Banach smooth are completely characterized. It is demonstrated that if $X$ is an $M$-embedded space then $X^*$ admits a predual which is not weakly Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ and also represents a split face, no subspace of $A(S)$ retains the property-$(wU)$ in $A(S)^{**}$. Furthermore, when $X=C_0(L)$, in the context of a locally compact Hausdorff space $L$, the continuity of the identity mapping $I:(B_{X^*},w^*)\to (B_{X^*},w)$ in $ext (B_{X^*})$ significantly influences the subspaces of $X$ that have unique extension property in $X^{**}$. Collectively, this study provides structural characterizations of specialized geometric property, so called Hahn-Banach smoothness, and offers solutions to some natural problems enlisted at the beginning that involve spaces that are $L_1$-preduals and also spaces that are $M$-embedded. |
| title | On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces |
| topic | Functional Analysis 46A22, 46B20, 46E15, 46B22, 46M05 |
| url | https://arxiv.org/abs/2601.01567 |