On the operator-valued Fourier transform of the Harish-Chandra Schwartz Algebra

Abstract

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 $0<p\le 2.$ This decomposition is then used to give an infinite-matrix-realization of the operator-valued Fourier image $$\mathfrak{F}:{\mathcal{C}}^p(G)\to {\mathcal{C}}^p(\hat{G})$$ 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 $$(({\mathfrak{F}}_B(\breve{\alpha }{)}_{(\gamma ,m)}(\Lambda )\otimes {\mathfrak{F}}_H(\breve{\alpha }{)}_{(\gamma ,l)}(Q:\chi :\nu ){)}_{\gamma \in F,(l,m)\in {\mathbb{Z}}^2}{)}_{F\subset \hat{K},|F|<\infty }$$ for every $\alpha \in {\mathcal{C}}^p(G).$ 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 $G.$

30 pages (including references). Comments are welcome