I am wondering whether a state $\omega$ on a $C^*$-algebra which is KMS (http://en.wikipedia.org/wiki/KMS_state) with respect to the group of automorphisms $\tau^t$, $t\in\mathbb{R}$, and at a given inverse temperature $\beta$ can be non-faithful. ($\omega$ is faithful iff $\omega(A^*A)=0$ for some element $A$ of the $C^*$-algebra implies $A=0$). Can anyone give me an example?
As a particular case: is there an example of a tracial state $\omega$ on a $C^*$-algebra which is not faithful? Tracial means that $\omega(AB)=\omega(BA)$ for all $A,B$, and since a tracial state is trivially KMS with respect to any group of automorphisms $\{\tau^t, t\in\mathbb{R}\}$, this would be an example of non-faithful KMS.
Every non-simple finite-dimensional C$^*$-algebra has non-faithful traces.
For the simplest case, let $A=\mathbb C\oplus\mathbb C$, $\varphi(a,b)=a$. Then $\varphi$ is tracial, non-faithful.
You can generalize the same idea to any direct sum of tracial algebras. If $A_1,A_2,\ldots$ are C$^*$-algebras with tracial states $\varphi_1,\varphi_2,\ldots$ respectively, then $$ \varphi(a_1,a_2,\ldots)=\varphi_1(a_1) $$ is a non-faithful tracial state on $\bigoplus_nA_n$. Another way to describe this situation is the case of $A$ with trace $\varphi$ and $p\in A$ a non-trivial central projection. Then $a\mapsto \varphi(pa)$ is a non-faithful trace.