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Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a $C^{*}$ algebraic structure.(That is, it has a $C^{*}$ involution).

I think that the most obvious obstruction for the disc algebra to be a $C^{*}$ algebra is that it does not have a zero divisor. So a related question ; Is it true to say that every Banach algebra cross norm on $A\otimes A$ gives us a Banach algebra without zero divisor?

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    $\begingroup$ The answer to your first question is no, because for any reasonable cross norm, the map $f\otimes g \to f(0)g$ will be a continuous algebra homomorphism of $A\otimes_\alpha A$ onto $A$. Banach algebra quotients of (algebras isomorphic to) abelian $C^*$-algebras are themselves abelian $C^*$-algebras, and $A$ is not isomorphic as a Banach algebra to any $C(X)$, so $A\otimes_\alpha A$ cannot be isomorphic as a Banach algebra to $C(X)$. The second question might be more interesting. $\endgroup$
    – Yemon Choi
    Commented Apr 26, 2014 at 19:16
  • $\begingroup$ Dear @Yemon thank you very much. Your comment is very interesting. It is also strange because in this question I was (very indirectly), somehow motivated by the following question:mathoverflow.net/questions/164169/…. on the other hand your map is $\phi \otimes Id$ in the above question. First Does your statement a consequence of the fact that the ideal in $C^{*}$ algebras are $*$-closed? second: Do you think that the tube lemma is true forthe disc algebra? Is the projective tensor product a reasonable norm? Thanks again for interesting comment. $\endgroup$
    – Ali Taghavi
    Commented Apr 26, 2014 at 19:34
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    $\begingroup$ Ali: 1) yes 2) don't know 3) yes, and for the disc algebra, the injective tensor product also gives you a Banach algebra. See Tomiyama's 1960 paper projecteuclid.org/euclid.tmj/1178244494 for more on tensor products of commutative Banach algebras $\endgroup$
    – Yemon Choi
    Commented Apr 26, 2014 at 19:45

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