Spectral properties and stability of perturbed Cartesian product

Abstract

Let $${\mathcal {A}}$$ and $${\mathcal {B}}$$ be commutative Banach algebras, and let $$T:{\mathcal {B}} \rightarrow {\mathcal {A}}$$ be an algebra homomorphism with $${\Vert T\Vert }\le 1$$ . Then T induces a Banach algebra product $$\times _T$$ perturbing the coordinatewise product on the Cartesian product space $${\mathcal {A}} \times {\mathcal {B}}$$ . We show that the spectral properties like spectral extension property, unique uniform norm property, regularity, weak regularity as well as Ditkin’s condition are stable with respect to this product.