Does there exist a $C^\*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?


I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no tracial states.



center of a $C^*$-algebra
There are lots of AF simple nonunital C$^\*$-algebras with (finite) traces (these have trivial centre). For example, begin with the usual $2^{\infty}$ UHF algebra, call it $B$, and take an infinite strictly increasing sequence of projections, $(p\_n)$, and form $\cup\_n p\_n B p\_n$; let $A$ be its closure. Then $A$ is simple (easy to check); it is nonunital (very easy) [and being simple, is thus centreless]; it has unique trace (given by the restriction of the unique trace on $B$) (easy). $A$ is of course a hereditary subalgebra of $B$, and there are hordes of similar examples.


If we set $\alpha$ to be the limit of the traces of the $p\_n$, then there is a significant difference (certainly wrt classification) between the algebras we obtain when $\alpha$ is rational, and when $\alpha $ is irrational. 


