Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^*c=cb^*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z_w(A)= \{ v \in A : av^*c=cv^*a \;\forall a,c \in A \}.$$

Are these notions of weak commutativity and weak center the same as the usual notions of commutativity and center in $C^{\ast}$-algebras?

By using approximate identity, I have managed to prove that both notions weakly commutative and commutative are same. It is also clear that center of $A$, i.e., $Z(A)$, contains $Z_w(A)$ but reverse inclusion is not clear. Any ideas?

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