  In this paper, we study the group Fourier transform and the Kohn-Nirenberg
quantization for homogeneous Lie groups as mappings between certain Gelfand
triples. For this, we restrict our considerations to the case, where the
homogeneous Lie group $G$ admits irreducible unitary representations, that are
square integrable modulo the center $Z(G)$ of $G$, and where $\dim Z(G)=1$.
Replacing the Schwartz space by a certain subspace $\mathcal S_*(G)
\hookrightarrow \mathcal S(G)$, we characterise the range of the group Fourier
transform on $\mathcal S_*(G)$ and construct distributions and Gelfand triples
around $L^2(G,\mu)$ and its Fourier image $L^2(\hat G,\hat \mu)$, such that the
Fourier transform becomes a Gelfand triple isomorphism. We give results on the
multiplication of distributions with a large class of vector valued smooth
functions and use this to establish the Kohn-Nirenberg quantization as an
isomorphism for our Gelfand triples and provide an explicit formula for the
Kohn-Nirenberg symbol of an operator.
