On the monotone $C^*$-algebra

TL;DR

This paper characterizes the concrete monotone $C^*$-algebra generated by Bernoulli-type monotone independent variables, showing it is a UHF algebra with an explicitly described Bratteli diagram and computable $K$-theory.

Contribution

It provides an abstract characterization of the monotone $C^*$-algebra, proves it is a UHF algebra, and explicitly describes its Bratteli diagram and $K$-theory.

Findings

The monotone $C^*$-algebra is a UHF algebra.

An explicit Bratteli diagram for the algebra is given.

The $K$-theory of the algebra is computed.

Abstract

The concrete monotone $C^*$-algebra, that is the (unital) $C^*$-algebra generated by monotone independent algebraic random variables of Bernoulli type, is characterized abstractly in terms of generators and relations and is shown to be UHF. Moreover, its Bratteli diagram is explicitly given, which allows for the computation of its $K$-theory.