Spectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times\mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta(d)a+\theta(b)c,bd)$ for all $(a,b),(c,d)\in\mathcal{A}\times\mathcal{B}$ . We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on $\theta$ .