Traces on separable simple $C^{\ast}$- algebras What is an example of a separable, simple $C^{\ast}$-algebra that admits two different tracial states?
EDIT: Julien has pointed to a number of avenues to answer this question. If anyone has an electronic copy of the paper of Longo he links to in the comment below, please post a summary of the argument. It would be nice to have access to a nice simple construction as advertised in the abstract of that paper.
 A: Blackadar showed [Traces on simple AF C*-algebras. J. Funct. Anal. 38 (1980), no. 2, 156–168] that every metrizable Choquet simplex arises as the trace space of a simple, AF algebra. 
A: Longo's example is as follows: he considers a C$^*$-dynamical system $(A,G,\alpha)$ with $A$ unital, simple, and with unique trace $\tau$; and $G$ discrete abelian. 
He notes that in this situation $\tau$ is $\alpha$-invariant (by the uniqueness of the trace) and thus $\alpha$ extends to $\bar\alpha:G\to\mbox{Aut}(\pi_\tau(A)'')$ (I didn't think why this is true).
And then he considers an additional condition on the action of the group. He requires the existence of nonzero $t\in G$ such that $\alpha_t$ is not inner, but $\bar\alpha_t=v\cdot v^*$, where $v\in\pi_\tau(A)''$ is an $\bar\alpha$-invariant unitary. 
Under those conditions, he proves that $A\rtimes_\alpha G$ is simple and has at least two traces. 
As a concrete example, he considers $A=\bigotimes_{n=1}^\infty\,M_{2^n}(\mathbb C)$, $G=\mathbb Z_2$, and the nontrivial element of $\mathbb Z_2$ given by conjugation by $\bigotimes_nu_n$, where $u_n$ is the diagonal matrix on $M_{2^n}(\mathbb C)$ that has diagonal $1,\ldots,1,-1$. 
