  This paper further investigates the implications of quasinilpotent
equivalence for (pairs of) elements belonging to the socle of a semisimple
Banach algebra. Specifically, not only does quasinilpotent equivalence of two
socle elements imply spectral equality, but also the trace, determinant and
spectral multiplicities of the elements must agree. It is hence shown that
quasinilpotent equivalence is established by a weaker formula (than that of the
spectral semidistance). More generally, in the second part, we show that two
elements possessing finite spectra are quasinilpotent equivalent if and only if
they share the same set of Riesz projections. This is then used to obtain
further characterizations in a number of general, as well as more specific
situations. Thirdly, we show that the ideas in the preceding sections turn out
to be useful in the case of $C^*$-algebras, but now for elements with infinite
spectra; we give two results which may indicate a direction for further
research.
