commutant of a non unital $C^*$ algebra

Question

If $A$ is a non-unital $C^*$ algebra,is the commutant of $A$ empty?

Does there exist a theorem which states that every $C^*$ algebra has commutant(centralizer)?

Answer 1

The centre $Z(A)$ of $A$ is never empty, because it always contains $0$.

It can be just $\{0\}$, though. Typical example would be $K(H)$. Since $K(H)'=\mathbb C I$, then $$ Z(K(H))=K(H)\cap K(H)'=\{0\}. $$