Uniqueness of Norm and Faithfulness of some Product Banach Algebras

Abstract: We prove that the faithful and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra. Further, we exhibit that these properties are not stable in null product algebra, and also give a common sufficient condition in terms of algebra norm for the co-dimension of $\mathcal{A}^2 = \text{span} { ab : a,b \in \mathcal{A}}$ to be finite in $\mathcal{A}$ and $\mathcal{A}^{2} = \mathcal{A} \ ( \text{when } \overline{\mathcal{A}^2} = \mathcal{A})$.