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?