  As a follow-up to a paper of D. Petz and J. Zem\'anek [4], a number of
equivalent conditions which characterize the trace among linear functionals on
matrix algebras, finite rank operators and the socle elements of semisimple
Banach algebras in general are given. Moreover, the converse problem is also
addressed, that is, given the equivalence of certain conditions which
characterize the trace, what can be said about the structure of the socle? In
particular, we characterize those socles isomorphic to matrix algebras in this
manner, as well as those socles which are minimal two-sided ideals.
