Let $A$ is a $C^*$-algebra and $\mu \in A^{**}$. Suppose that $a \mu = \mu a$ for all $a \in A$. Then $\mu \in Z(A^{**})$.
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1$\begingroup$ $\nu = \mu$???? $\endgroup$– David HandelmanCommented Dec 4, 2016 at 15:24
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1$\begingroup$ If you are the same person who has been asking about the derivation problem for Cstar algebras (which, incidentally, predates the derivation problem for group algebras, and which was one of Johnson's main motivations for studying the derivation problem for group algebras) then I suggest you get your accounts merged. It does not help anyone if you have several anonymous accounts and it may create suspicions of sockpuppetry $\endgroup$– Yemon ChoiCommented Dec 4, 2016 at 19:18
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1$\begingroup$ I'm voting to close this question because the OP is not engaging with the responses and has not clarified the context for their question $\endgroup$– Yemon ChoiCommented Dec 5, 2016 at 20:26
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1 Answer
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I think this follows from the fact that multiplication in $A^{**}$ is separately weak-star continuous, i.e.
for all $a\in A^{**}$ the function $A^{**} \to A^{**}$, $b\mapsto ab$, is weak-star to weak-star continuous;
for all $a\in A^{**}$ the function $A^{**} \to A^{**}$, $b\mapsto ba$, is weak-star to weak-star continuous.
Therefore this should work for any Arens regular Banach algebra $A$.
answered Dec 4, 2016 at 15:50
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$\begingroup$ For a Cstar algebra, appealing to Arens regularity may be seen as excessively abstract: just use results about von Neumann algebras (represented concretely on B(H) ) and the various topologies on them $\endgroup$ Commented Dec 4, 2016 at 17:00
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$\begingroup$ @user347731 Dear anonymous user, that was a typo. Apologies. $\endgroup$ Commented Dec 4, 2016 at 19:13