Polynomial growth and functional calculus in algebras of integrable cross-sections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929758745395200 |
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| author | Flores, Felipe I. |
| author_facet | Flores, Felipe I. |
| contents | Let ${\sf G}$ be a locally compact group with polynomial growth of order $d$, a polynomial weight $ν$ on ${\sf G}$ and a Fell bundle $\mathscr C\overset{q}{\to}{\sf G}$. We study the Banach $^*$-algebras $L^1({\sf G}\,\vert\,\mathscr C)$ and $L^{1,ν}({\sf G}\,\vert\,\mathscr C)$, consisting of integrable cross-sections with respect to ${\rm d} x$ and $ν(x){\rm d} x$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^*$-module $L^2_{\rm e}({\sf G}\,\vert\,\mathscr C)$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy $$\|{e^{itΦ}}\|=O(|t|^n),\quad\text{ as }|t|\to\infty,$$ for values of $n$ that only depend on $d$ and the weight $ν$. We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, $^*$-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with $C^*$-dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial growth and functional calculus in algebras of integrable cross-sections Flores, Felipe I. Functional Analysis Operator Algebras Primary 43A20, Secondary 47L65, 47L30 Let ${\sf G}$ be a locally compact group with polynomial growth of order $d$, a polynomial weight $ν$ on ${\sf G}$ and a Fell bundle $\mathscr C\overset{q}{\to}{\sf G}$. We study the Banach $^*$-algebras $L^1({\sf G}\,\vert\,\mathscr C)$ and $L^{1,ν}({\sf G}\,\vert\,\mathscr C)$, consisting of integrable cross-sections with respect to ${\rm d} x$ and $ν(x){\rm d} x$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^*$-module $L^2_{\rm e}({\sf G}\,\vert\,\mathscr C)$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy $$\|{e^{itΦ}}\|=O(|t|^n),\quad\text{ as }|t|\to\infty,$$ for values of $n$ that only depend on $d$ and the weight $ν$. We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, $^*$-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with $C^*$-dynamical systems. |
| title | Polynomial growth and functional calculus in algebras of integrable cross-sections |
| topic | Functional Analysis Operator Algebras Primary 43A20, Secondary 47L65, 47L30 |
| url | https://arxiv.org/abs/2401.09730 |