On Trace Characterizations and Socle Identifications in Ordered Banach Algebras

As a follow-up to D. Petz and J. Zemánek [1], and the methods of the authors in [7] we show a number of equivalent conditions which characterize the trace among linear functionals on ordered matrix algebras, finite rank operators and the ordered socle elements of semi simple, Banach algebras in general [7]. Investigate the converse problem; that is, given the equivalence of certain conditions which characterize the trace, what can be said about the structure of the socle? Hence, we characterize those socles isomorphic to ordered matrix algebras, as well as those ordered socles which are minimal two-sided ideals.