Random amenable $\mathrm{C}^*$-algebras

TL;DR

This paper explores probabilistic properties of various classes of random $ ext{C}^*$-algebras, providing computable answers to questions about their structure, stability, and invariants through graph-based models.

Contribution

It introduces a framework for analyzing the likelihood of structural features in random $ ext{C}^*$-algebras using graph-based constructions, offering new insights into their probabilistic behavior.

Findings

Probability of infinite type UHF algebras

Distribution of extremal traces in AI algebras

Expected radius of comparison in Villadsen-type AH algebras

Abstract

What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most $k$ extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz-Krieger algebra is purely infinite and simple, and what can be said about the distribution of its $K$-theory? By constructing $\mathrm{C}^*$-algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.