  In this work we establish the metric approximation property for Besov spaces
defined on arbitrary compact Lie groups. As a consequence of this fact, we
investigate trace formulae for nuclear Fourier multipliers on Besov spaces.
Finally, we study the r-nuclearity, the Grothendieck-Lidskii formula and the
(nuclear) trace of pseudo-differential operators in generalized H\"ormander
classes acting on periodic Besov spaces. We will restrict our attention to
pseudo-differential operators with symbols of limited regularity.
