tracial states on $C^*$ subalgebra

Question

Let $A$ be a $C^*$ algebra,$A=A_1\oplus A_2$.If $A$ has tracial state $\tau$,I want to show $A_1$ also has tracial state,say $\tau_1$.

My thought: let $\tau_1(a_1)=\tau(a_1,0)$,where $a_1 \in A_1$,but this construction does not assure that $\tau_1$ is nonzero. Does there exist tracial state on $A_1$?

Answer 1

No. Take $A=B(H)\oplus \mathbb C$, with $\tau(T,\lambda)=\lambda$. Here $A_1$ has no tracial state.