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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.20755 |
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| _version_ | 1866929442712977408 |
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| author | Oyadare, Olufemi O. |
| author_facet | Oyadare, Olufemi O. |
| contents | We establish a $K-$type decomposition of the Harish-Chandra Schwartz algebra $\mathcal{C}^{p}(G),$ for any real-rank $1$ reductive group $G$ with a maximal compact subgroup $K$ and $0<p\leq2.$ This decomposition is then used to give an infinite-matrix-realization of the operator-valued Fourier image $$\mathfrak{F}:\mathcal{C}^{p}(G)\rightarrow \mathcal{C}^{p}(\hat{G})$$ of $\mathcal{C}^{p}(G)$ as a Fr$\acute{e}$chet multiplication algebra in which every member of $\mathcal{C}^{p}(\hat{G})$ consists of a countable block-matrices of the form $$((\mathfrak{F}_{B}(\breveα)_{(γ,m)}(Λ)\otimes\mathfrak{F}_{H}(\breveα)_{(γ,l)}(Q:χ:ν))_{γ\in F, (l,m)\in\mathbb{Z}^{2}})_{F\subset \hat{K},|F|<\infty}$$ for every $α\in \mathcal{C}^{p}(G).$ This proves Trombi's conjecture for $G$ of real rank $1$ and the technique leads to a proof of the fundamental theorem of harmonic analysis for any arbitrary real-rank reductive group $G.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the operator-valued Fourier transform of the Harish-Chandra Schwartz Algebra Oyadare, Olufemi O. Representation Theory Functional Analysis 46A04, 43A15, 22E46 We establish a $K-$type decomposition of the Harish-Chandra Schwartz algebra $\mathcal{C}^{p}(G),$ for any real-rank $1$ reductive group $G$ with a maximal compact subgroup $K$ and $0<p\leq2.$ This decomposition is then used to give an infinite-matrix-realization of the operator-valued Fourier image $$\mathfrak{F}:\mathcal{C}^{p}(G)\rightarrow \mathcal{C}^{p}(\hat{G})$$ of $\mathcal{C}^{p}(G)$ as a Fr$\acute{e}$chet multiplication algebra in which every member of $\mathcal{C}^{p}(\hat{G})$ consists of a countable block-matrices of the form $$((\mathfrak{F}_{B}(\breveα)_{(γ,m)}(Λ)\otimes\mathfrak{F}_{H}(\breveα)_{(γ,l)}(Q:χ:ν))_{γ\in F, (l,m)\in\mathbb{Z}^{2}})_{F\subset \hat{K},|F|<\infty}$$ for every $α\in \mathcal{C}^{p}(G).$ This proves Trombi's conjecture for $G$ of real rank $1$ and the technique leads to a proof of the fundamental theorem of harmonic analysis for any arbitrary real-rank reductive group $G.$ |
| title | On the operator-valued Fourier transform of the Harish-Chandra Schwartz Algebra |
| topic | Representation Theory Functional Analysis 46A04, 43A15, 22E46 |
| url | https://arxiv.org/abs/2407.20755 |