A Banach algebra A is without order if for all $x \in A$, $xA=\{0\}$ implies $x=0$, or, for all $x \in A$, $Ax=\{0\}$ implies $x=0$. If $G$ is a compact abelian group then $L_{\infty}(G)$ is a Banach algebra with the convolution product and its second dual is a Banach algebra with the first Arens product. I like know if $L_{\infty}(G)^{**}$ is a without order Banach algebra.
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One thing which might help with calculations is the following fact: if $A$ is any Banach algebra whose underlying Banach space is isomorphic to $C(K)$ for some $K$, then $A$ is Arens regular. (This works by using the WAP characterization of Arens regularity that is due, I think, to John Pym, together with the fact that every bounded linear map $C(K)\to C(K)^*$ is weakly compact.) – Yemon Choi Jun 03 '16 at 02:56
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Would you mind if I mentioned this question to my colleague Garth Dales? If you are a PhD student or a postdoc, and would prefer to keep this problem for yourself to work on, then let me know; I am sure Garth would respect that preference. If you are happy discussing the question non-pseudonymously, you could even contact him yourself – Yemon Choi Jun 03 '16 at 13:29
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I thank you for your indications. I don't have problem with that you consult your colleague Garth Dales. In fact I would be very grateful. I send an email to you, but I have received no reply. @YemonChoi – El nota Jun 08 '16 at 23:08
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I got your email, sorry for not replying. Garth and I have both been rather busy; I will pass this question on to him – Yemon Choi Jun 08 '16 at 23:54