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