Intereting Posts

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Universal property of tensor products / Categorial expression of bilinearity
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Explicit construction of a initial object in a topos
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The purpose of the $\sf ZFC$ Axiom of Infinity

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.

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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.

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