Weak$^*$-weak points of continuity on the state spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910134338322432 |
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| author | Dwivedi, Saurabh |
| author_facet | Dwivedi, Saurabh |
| contents | Let $X$ be a Banach space. For $x \in X$ with $\|x\| = 1$, we denote the state space by $S_x = \{x^* \in X^* : \|x^*\| = x^*(x) = 1\}.$ In this paper, we study weak$^*$-weak and weak$^*$-$\|\cdot\|$ points of continuity of the identity map on the state spaces in the space $\ell^p(X)$ for $1 < p < \infty$, where $X$ is a non-reflexive Banach space. We then use these results to characterize the weak and norm compactness of the state spaces of unit vectors in $\ell^p(X)$. In addition, we address an open problem concerning the characterization of weakly compact state spaces in the space of Bochner-integrable functions $L^1(μ, X)$. We also provide a local solution to this problem without any additional assumptions on the Banach space $X$. Motivated by the work of S. Daptari, V. Montesinos, and T. S. S. R. K. Rao, we show that if the set of all weak$^*$-weak points of continuity of $L^1(μ, X)_1^*$ is weakly dense in $L^1(μ, X)_1^*$, then $X^*$ has the Radon-Nikodým property (RNP). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08732 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak$^*$-weak points of continuity on the state spaces Dwivedi, Saurabh Functional Analysis Primary 46A22, 46B10, 46B25, Secondary 46B20, 46B22 Let $X$ be a Banach space. For $x \in X$ with $\|x\| = 1$, we denote the state space by $S_x = \{x^* \in X^* : \|x^*\| = x^*(x) = 1\}.$ In this paper, we study weak$^*$-weak and weak$^*$-$\|\cdot\|$ points of continuity of the identity map on the state spaces in the space $\ell^p(X)$ for $1 < p < \infty$, where $X$ is a non-reflexive Banach space. We then use these results to characterize the weak and norm compactness of the state spaces of unit vectors in $\ell^p(X)$. In addition, we address an open problem concerning the characterization of weakly compact state spaces in the space of Bochner-integrable functions $L^1(μ, X)$. We also provide a local solution to this problem without any additional assumptions on the Banach space $X$. Motivated by the work of S. Daptari, V. Montesinos, and T. S. S. R. K. Rao, we show that if the set of all weak$^*$-weak points of continuity of $L^1(μ, X)_1^*$ is weakly dense in $L^1(μ, X)_1^*$, then $X^*$ has the Radon-Nikodým property (RNP). |
| title | Weak$^*$-weak points of continuity on the state spaces |
| topic | Functional Analysis Primary 46A22, 46B10, 46B25, Secondary 46B20, 46B22 |
| url | https://arxiv.org/abs/2604.08732 |