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Let $A$ be a $C^*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?

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up vote 5 down vote accepted

No. Take $A$ to be $c$, the space of convergent sequences. Its second dual is $l^\infty$, which is atomic, but it is not an ideal of $l^\infty$.

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Or more generally, let $K$ be a compact Hausdorff space which is countable infinite. Then $C(K)^*=\ell^1(K)$ and so $C(K)^{**}=\ell^\infty(K)$, but $C(K)$ is unital, so won't be an ideal in its bidual. –  Matthew Daws Apr 25 '12 at 6:56
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