Multiplier algebras of $L^p$-operator algebras
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| Format: | Preprint |
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2024
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| _version_ | 1866915085605142528 |
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| author | Blinov, Andrey Delfín, Alonso Weld, Ellen |
| author_facet | Blinov, Andrey Delfín, Alonso Weld, Ellen |
| contents | It is known that the multiplier algebra of an approximately unital and nondegenerate $L^p$-operator algebra is again an $L^p$-operator algebra. In this paper we investigate examples that drop both hypotheses. In particular, we show that the multiplier algebra of $T_2^p$, the algebra of strictly upper triangular $2 \times 2$ matrices acting on $\ell_2^p$, is still an $L^p$-operator algebra for any $p$. To contrast this result, we first provide a thorough study of the augmentation ideal of $\ell^1(G)$ for a discrete group $G$. We use this ideal to define a family of nonapproximately unital degenerate $L^p$-operator algebras, $F_{0}^p(\Bbb{Z}/3\Bbb{Z})$, whose multiplier algebras cannot be represented on any $L^q$-space for any $q \in [1, \infty)$ as long as $p \in [1, p_0] \cup [p_0', \infty)$, where $p_0=1.606$ and $p_0'$ is its Hölder conjugate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplier algebras of $L^p$-operator algebras Blinov, Andrey Delfín, Alonso Weld, Ellen Functional Analysis Operator Algebras Primary 46H15, 46H35, Secondary 47L10 It is known that the multiplier algebra of an approximately unital and nondegenerate $L^p$-operator algebra is again an $L^p$-operator algebra. In this paper we investigate examples that drop both hypotheses. In particular, we show that the multiplier algebra of $T_2^p$, the algebra of strictly upper triangular $2 \times 2$ matrices acting on $\ell_2^p$, is still an $L^p$-operator algebra for any $p$. To contrast this result, we first provide a thorough study of the augmentation ideal of $\ell^1(G)$ for a discrete group $G$. We use this ideal to define a family of nonapproximately unital degenerate $L^p$-operator algebras, $F_{0}^p(\Bbb{Z}/3\Bbb{Z})$, whose multiplier algebras cannot be represented on any $L^q$-space for any $q \in [1, \infty)$ as long as $p \in [1, p_0] \cup [p_0', \infty)$, where $p_0=1.606$ and $p_0'$ is its Hölder conjugate. |
| title | Multiplier algebras of $L^p$-operator algebras |
| topic | Functional Analysis Operator Algebras Primary 46H15, 46H35, Secondary 47L10 |
| url | https://arxiv.org/abs/2403.16309 |