On the continuity of intertwining operators over generalized convolution algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929529795117056 |
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| author | Flores, Felipe I. |
| author_facet | Flores, Felipe I. |
| contents | Let ${\sf G}$ be a locally compact group, $\mathscr C\overset{q}{\to}{\sf G}$ a Fell bundle and $\mathfrak B=L^1({\sf G}\,\vert\,\mathscr C)$ the algebra of integrable cross-sections associated to the bundle. We give conditions that guarantee the automatic continuity of an intertwining operator $θ:\mathcal X_1\to\mathcal X_2$, where $\mathcal X_1$ is a Banach $\mathfrak B$-bimodule and $\mathcal X_2$ is a weak Banach $\mathfrak B$-bimodule, in terms of the continuity ideal of $θ$. We provide examples of algebras where this conditions are met, both in the case of derivations and algebra morphisms. In particular, we show that, if ${\sf G}$ is infinite, finitely-generated, has polynomial growth and $α$ is a free (partial) action of ${\sf G}$ on the compact space $X$, then every homomorphism of $\ell^1_α({\sf G},C(X))$ into a Banach algebra is automatically continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the continuity of intertwining operators over generalized convolution algebras Flores, Felipe I. Functional Analysis Operator Algebras Primary 43A20, Secondary 47L65, 46H40 Let ${\sf G}$ be a locally compact group, $\mathscr C\overset{q}{\to}{\sf G}$ a Fell bundle and $\mathfrak B=L^1({\sf G}\,\vert\,\mathscr C)$ the algebra of integrable cross-sections associated to the bundle. We give conditions that guarantee the automatic continuity of an intertwining operator $θ:\mathcal X_1\to\mathcal X_2$, where $\mathcal X_1$ is a Banach $\mathfrak B$-bimodule and $\mathcal X_2$ is a weak Banach $\mathfrak B$-bimodule, in terms of the continuity ideal of $θ$. We provide examples of algebras where this conditions are met, both in the case of derivations and algebra morphisms. In particular, we show that, if ${\sf G}$ is infinite, finitely-generated, has polynomial growth and $α$ is a free (partial) action of ${\sf G}$ on the compact space $X$, then every homomorphism of $\ell^1_α({\sf G},C(X))$ into a Banach algebra is automatically continuous. |
| title | On the continuity of intertwining operators over generalized convolution algebras |
| topic | Functional Analysis Operator Algebras Primary 43A20, Secondary 47L65, 46H40 |
| url | https://arxiv.org/abs/2403.11039 |