Consider \(\mathcal {A}\) , a commutative Banach algebra, and \(\mathcal {I}\) , a closed ideal in \(\mathcal {A}\) . Then \(\mathcal {A} \times \mathcal {I}\) is a commutative Banach algebra with linear operations, multiplication defined as \((a,x)(b,y) = (ab - xy, ay + bx)\) and the norm \(\Vert (a,x)\Vert = \Vert a\Vert + \Vert x\Vert \) for all \((a, x) \in \mathcal {A} \times \mathcal {I}\) . This particular Banach algebra is denoted as \(\mathcal {A} \times _z \mathcal {I}\) , and we refer to it as the complex product Banach algebra. In the case where \(\mathcal {I}\) is a spectral synthesis ideal in \(\mathcal {A}\) , we establish a critical equivalence: \(\mathcal {A} \times _z \mathcal {I}\) is a BSE-algebra if and only if \(\mathcal {A}\) itself is a BSE-algebra. Similarly, the algebra \(\mathcal {A} \times _z \mathcal {I}\) qualifies as a BED-algebra precisely when \(\mathcal {A}\) is a BED-algebra.
