Trace Characterizations and Socle Identifications in Banach Algebras

TL;DR

This paper explores conditions that characterize the trace in Banach algebras and investigates the structure of socles, including their relation to matrix algebras and minimal ideals.

Contribution

It provides new equivalent conditions characterizing the trace and analyzes the structure of socles in semisimple Banach algebras.

Findings

Characterization of trace via equivalent conditions

Identification of socles isomorphic to matrix algebras

Description of socles as minimal two-sided ideals

Abstract

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.