Geometry of the unit ball of ${\mathcal L}(X,Y^*)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908642296463360 |
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| author | Rao, T. S. S. R. K. Seal, Susmita |
| author_facet | Rao, T. S. S. R. K. Seal, Susmita |
| contents | In this work we study the geometry of the unit ball of the space of operators ${\mathcal L}(X,Y^*)$, by considering the projective tensor product $X\hat{\otimes}_π Y$ as a predual. We prove that if an elementary tensor (rank one operator) of the form $x_0^*\otimes y_0^* $ in the unit sphere $ S_{{\mathcal L}(X,Y^*)}$ is a weak$^*$-strongly extreme point of the unit ball, then $x_0^*$ is weak$^*$-strongly extreme point of unit ball of $X^*$ and $y_0^*$ is weak$^*$-strongly extreme point of the unit ball of $Y^*$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak$^*$-weak continuity for the identity mapping) on the unit sphere of ${\mathcal L}(X,Y^*)$. We also study extremal phenomenon in the unit ball of ${\mathcal L}(X,Y^*)^*$. We partly solve the open problem, when does an elementary tensor, whose components are Namioka points is again a Namioka point? We show that if a point $z\in S_{{\mathcal L}(X,Y^*)^*}$ is a weak$^*$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak$^*$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^*)$ is separating for $X\hat{\otimes}_π Y$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of the unit ball of ${\mathcal L}(X,Y^*)$ Rao, T. S. S. R. K. Seal, Susmita Functional Analysis 46A22, 46B10, 46B25, 46B22, 47L05 In this work we study the geometry of the unit ball of the space of operators ${\mathcal L}(X,Y^*)$, by considering the projective tensor product $X\hat{\otimes}_π Y$ as a predual. We prove that if an elementary tensor (rank one operator) of the form $x_0^*\otimes y_0^* $ in the unit sphere $ S_{{\mathcal L}(X,Y^*)}$ is a weak$^*$-strongly extreme point of the unit ball, then $x_0^*$ is weak$^*$-strongly extreme point of unit ball of $X^*$ and $y_0^*$ is weak$^*$-strongly extreme point of the unit ball of $Y^*$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak$^*$-weak continuity for the identity mapping) on the unit sphere of ${\mathcal L}(X,Y^*)$. We also study extremal phenomenon in the unit ball of ${\mathcal L}(X,Y^*)^*$. We partly solve the open problem, when does an elementary tensor, whose components are Namioka points is again a Namioka point? We show that if a point $z\in S_{{\mathcal L}(X,Y^*)^*}$ is a weak$^*$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak$^*$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^*)$ is separating for $X\hat{\otimes}_π Y$. |
| title | Geometry of the unit ball of ${\mathcal L}(X,Y^*)$ |
| topic | Functional Analysis 46A22, 46B10, 46B25, 46B22, 47L05 |
| url | https://arxiv.org/abs/2501.15783 |