Non-linear traces of Choquet type on AF algebras

TL;DR

This paper extends the concept of non-linear Choquet-type traces from matrix algebras to all unital AF algebras, establishing a correspondence with increasing functions on the dimension scale and providing explicit formulas.

Contribution

It generalizes the characterization of Choquet traces to arbitrary unital AF algebras and links them to functions on the dimension scale.

Findings

One-to-one correspondence between Choquet traces and increasing functions on the dimension scale.

Explicit formulas for Choquet traces in terms of spectrum and spectral projections.

Extension of Nagisa--Watatani's characterization to a broader class of algebras.

Abstract

We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale, and we obtain explicit Choquet formulas in terms of the spectrum and ranks of spectral projections along a fixed AF filtration.