Non-existence Tracial states

Question

We know that every non-zero finite dimensional C*-algebra has a tracial state. I am searching for an example of a simple C* algebra without tracial state with explaination. I think you have to look to the calkin algebra on a separable Hilbert space. Someone an idea? Thanks.

Answer 1

Consider the Cuntz algebra $\mathcal O_2$. This is a simple, purely infinite C*-algebra. This means that there exist pairwise orthogonal projections $p,q\in\mathcal O_2$ with $1\sim p\sim q$. Now suppose that $f$ is a tracial state on $\mathcal O_2$.

Then $$ 1=f(1)=f(p)=f(q). $$ So $$ 0\leq f(1-(p+q))=f(1)-f(p)-f(q)=-1, $$ a contradiction.