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

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 $$((\mathfrak{F}_{B}(\breve{\alpha})_{(\gamma,m)}(\Lambda)\otimes\mathfrak{F}_{H}(\breve{\alpha})_{(\gamma,l)}(Q:\chi:\nu))_{\gamma\in F, (l,m)\in\mathbb{Z}^{2}})_{F\subset \hat{K},|F|<\infty}$$ 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 $G.$