  We introduce an abstract theory of the principal symbol mapping for
pseudodifferential operators extending the results of a preceding paper and
providing a simple algebraic approach to the theory of pseudodifferential
operators in settings important in noncommutative geometry. We provide a
variant of Connes' trace theorem which applies to certain noncommutative
settings, with a minimum of technical preliminaries. Our approach allows us to
consider in a operators with non-smooth symbols, and we demonstrate the power
of our approach by extending Connes' trace theorem to operators with non-smooth
symbols in three examples: the Lie group $\mathrm{SU}(2)$, noncommutative tori
and Moyal planes.
