Suppose $A$ is a $C^*$ algebra ,$\phi$ is a tracial state (not faithful)on $A$.Can we construct a faithful tracial state on$A$ by using $\phi$?
2026-03-26 01:14:40.1774487680
construct a faithful tracial state throgh a tracial state on some $C^*$ algebra
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No. There are C$^*$-algebras that do not have tracial states at all. So, for instance, take $A=B(H)\oplus\mathbb C$, and let $\phi(a\oplus\lambda)=\lambda$. Then $\phi$ is a tracial state, but $A$ has no faithful tracial state.