trace on quotient of a $C^*$ algebra

Question

Does there exist a $C^*$-algebra $A$ such that $A$ has a faithful tracial state,but the quotient of $A$ has no tracial states(there exists a nontrivial ideal $I$ of $A$ such that $A/I$ has no traces )

Answer 1

Sure, take $$ A := \mathbb C \oplus \mathcal O_2, $$ where $\mathcal O_2$ is the Cuntz-algebra. Define $\tau_A(\lambda,x) := \lambda$. Then $\tau_A$ is a tracial state of $A$ but $A / \mathbb C \cong \mathcal O_2$ which admits no tracial states.

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For a faithful tracial state, consider $A := M(\mathcal W)$, where $\mathcal W$ is the non-unital stably projectionless separable simple nuclear C*-algebra which admits a unique tracial state and is KK-equivalent to $0$, called the Jacelon-Razak algebra. By [1] the Corona C*-algebra $M(\mathcal W) / \mathcal W$ is purely infinite, hence has no tracial states, however $M(\mathcal W)$ has a faithful tracial state, namely the extension of the unique tracial state on $\mathcal W$.

To see that the extension is faithful, assume $\tau(x) = 0$ where $x \in M(\mathcal W)_+$. Then $\tau(e_\lambda x e_\lambda) = 0$ where $(e_\lambda)$ is an approximate identity for $\mathcal W$ and hence $e_\lambda x e_\lambda = 0$ since $\tau$ is faithful on $\mathcal W$. It follows that $x = 0$ since $e_\lambda \to 1$ strictly.

[1] Lin, Huaxin, Simple corona $C^*$-algebras, Proc. Am. Math. Soc. 132, No. 11, 3215-3224 (2004). ZBL1049.46040.