  To a Boolean inverse monoid $S$ we associate a universal C*-algebra
$C_B^*(S)$ and show that it is equal to Exel's tight C*-algebra of $S$. We then
show that any invariant mean on $S$ (in the sense of Kudryavtseva, Lawson, Lenz
and Resende) gives rise to a trace on $C_B^*(S)$, and vice-versa, under a
condition on $S$ equivalent to the underlying groupoid being Hausdorff. Under
certain mild conditions, the space of traces of $C_B^*(S)$ is shown to be
isomorphic to the space of invariant means of $S$. We then use many known
results about traces of C*-algebras to draw conclusions about invariant means
on Boolean inverse monoids; in particular we quote a result of Blackadar to
show that any metrizable Choquet simplex arises as the space of invariant means
for some AF inverse monoid $S$.
