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

We establish a $K-$type decomposition of the Harish-Chandra Schwartz algebra Cᵖ(G), for any real-rank $1$ reductive group G with a maximal compact subgroup K and 0<p≤2. This decomposition is then used to give an infinite-matrix-realization of the operator-valued Fourier image F:Cᵖ(G)→ Cᵖ(Ĝ) of Cᵖ(G) as a Frechet multiplication algebra in which every member of Cᵖ(Ĝ) consists of a countable block-matrices of the form ((FB(α)(γ,m)(Λ)H(α)(γ,l)(Q:χ:ν))_γ∈ F, (l,m)²)F⊂ K̂,|F|<∞ for every α∈ Cᵖ(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 $G.$