On the monotone $C^{\ast }$-algebra

The concrete monotone $C^{\ast }$-algebra, that is the (unital) $C^{\ast }$-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.