# Just-infinite C*-algebras and their invariants

## TL;DR

This paper explores the structure of just-infinite C*-algebras, showing that their trace simplices can represent any infinite-dimensional Choquet simplex, and provides a detailed analysis of their invariants and Bratteli diagrams.

## Contribution

It extends the construction of just-infinite residually finite dimensional C*-algebras to realize any infinite-dimensional Choquet simplex as a trace simplex.

## Key findings

- Any infinite-dimensional Choquet simplex can be realized as the trace simplex of a just-infinite residually finite dimensional C*-algebra.
- The trace simplex of certain AF-algebras has exactly one extremal trace of type II_1.
- A modification of Bratteli diagrams can produce just-infinite residually finite dimensional AF-algebras.

## Abstract

Just-infinite C*-algebras, i.e., infinite dimensional C*-algebras, whose proper quotients are finite dimensional, were investigated in [Grigorchuk-Musat-Rordam, 2016]. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in that article. In this paper we extend that construction by showing that each infinite dimensional metrizable Choquet simplex is affinely homeomorphic to the trace simplex of a just-infinite residually finite dimensional C*-algebras. The trace simplex of any unital residually finite dimensional C*-algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in the above mentioned article, and we show that it has precisely one extremal trace of type II_1. We give a complete description of the Bratteli diagrams corresponding to residually finite dimensional AF-algebras. We show that a modification of any such Bratteli diagram, similar to the modification that makes an arbitrary Bratteli diagram simple, will yield a just-infinite residually finite dimensional AF-algebra.