Sherman-Takeda type theorems for locally C*-algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915729354260480 |
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| author | Singh, Lav Kumar Peperko, Aljoša |
| author_facet | Singh, Lav Kumar Peperko, Aljoša |
| contents | In this article, we will first establish some density results for a locally $C^*$-algebra $\mathcal A$ and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous $*$-representation $φ$ of $\mathcal A^{**}$ (equipped with unique Arens product) on the space $B_{loc}(\mathcal H)$ such that $φ(\mathcal A^{**})\subset \overline{π(\mathcal A)}^{WOT}$, where $π:\mathcal A\to B_{loc}(\mathcal H)$ is the associated universal $*$-representation and $\mathcal H$ is the associated locally Hilbert space. Finally we show that for a Fréchet locally $C^*$-algebra $\mathcal A$ possessing KDP, the second strong dual is algebraically and topologically $*$-isomorphic to $ \overline{π(\mathcal A)}^{WOT}$, which is a direct analogue of the classical Sherman-Takeda theorem for $C^*$-algebras. We shall also observe the joint continuity of some associated bilinear maps in the running. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00717 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sherman-Takeda type theorems for locally C*-algebras Singh, Lav Kumar Peperko, Aljoša Operator Algebras 46H15, 46K10, 46K05, 46H35 In this article, we will first establish some density results for a locally $C^*$-algebra $\mathcal A$ and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous $*$-representation $φ$ of $\mathcal A^{**}$ (equipped with unique Arens product) on the space $B_{loc}(\mathcal H)$ such that $φ(\mathcal A^{**})\subset \overline{π(\mathcal A)}^{WOT}$, where $π:\mathcal A\to B_{loc}(\mathcal H)$ is the associated universal $*$-representation and $\mathcal H$ is the associated locally Hilbert space. Finally we show that for a Fréchet locally $C^*$-algebra $\mathcal A$ possessing KDP, the second strong dual is algebraically and topologically $*$-isomorphic to $ \overline{π(\mathcal A)}^{WOT}$, which is a direct analogue of the classical Sherman-Takeda theorem for $C^*$-algebras. We shall also observe the joint continuity of some associated bilinear maps in the running. |
| title | Sherman-Takeda type theorems for locally C*-algebras |
| topic | Operator Algebras 46H15, 46K10, 46K05, 46H35 |
| url | https://arxiv.org/abs/2601.00717 |