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 ${\mathcal{A}}^2=\mathcal{A}(\text{when }\overline{{\mathcal{A}}^2}=\mathcal{A})$.