  In a recent paper by M. Mantoiu and M. Ruzhansky, a global
pseudo-differential calculus has been developed for unimodular groups of type
I. In the present article we generalize the main results to arbitrary locally
compact groups of type I. Our methods involve the use of Plancherel's theorem
for non-unimodular groups. We also make connections with a $C^*$-algebraic
formalism, involving dynamical systems, and give explicit constructions for the
group of affine transformations of the real line.
