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Let $A$ and $B$ be $C^*$-algebras with centers $Z_A$ and $Z_B$. Suppose $\rho：A\rightarrow B$ is a surjective $*$- homomorphism. It is easy to check $\rho(Z_A)\subset Z(B)$.

I wonder how to assure that $\rho(Z_A) \neq Z(B)$?

If $rho$ is not surjective, what is the relationship between $\rho(Z_A)$ and $Z_B$?

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    $\begingroup$ I don't quite understand the wording of your question. Are you seeking an explicit example of $A$ and $B$ with a surjective star-homomorphism $\rho:A\to B$ for which $\rho(Z_A)\neq Z_B$? Or are you asking for some kind of characterization? I think I can produce a specific (counter)example, but I don't think there will be any sensible characterization of when the inclusion is strict. $\endgroup$
    – Yemon Choi
    Commented Feb 15, 2022 at 23:20
  • $\begingroup$ Whether there exists some characterization? In Jorgen Vesterstrom's peper " on the homomorphism image of the cebter of a $C^*$-algebra", the author mentions some characterization. But in that theorem, we need to know the concrete center, if we can calculate the center of $A$, how to determine whether $\rho(Z_A)$ is equal to $Z_B$? $\endgroup$
    – math112358
    Commented Feb 16, 2022 at 0:26

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