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 $G.$
Categories: math.RT math.FA