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 increasing family of projections $p_n$, and form $\cup p_n B p_n$; let $A$ be its closure. Then $A$ is simple (easy to check); it is nonunital (very easy); 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 we can obtain lots 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.

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.