Some Banach algebra properties in the Cartesian product of Banach algebras (Q376629)

The authors consider the Cartesian product \({\mathcal A}\times{\mathcal B}\) of two semisimple commutative Banach algebras \(\mathcal A\) and \(\mathcal B\). They show that the unique uniform norm property (UUNP), regularity, quasi divisor of zero property (QDZP) and topological annihilator property (TAP) are properties which \({\mathcal A}\times{\mathcal B}\) has if and only if both \(\mathcal A\) and \(\mathcal B\) have the same property. They also prove that if one drops the condition of commutativity on \(\mathcal A\) and \(\mathcal B\), then the spectral extension property (SEP) and the weak spectral extension property (WSEP) are such properties that, if \({\mathcal A}\times{\mathcal B}\) has one of these properties, then both \(\mathcal A\) and \(\mathcal B\) have the same property. It remains an open question whether the converse of the last claim is also true.