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Ali Taghavi
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tensor product of the disc algebra with itself

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?

Ali Taghavi
  • 356
  • 8
  • 31
  • 123