A Class of algebras admitting infinitely many norm topologies

Let $\mathcal{A}$ be an algebra, and let ${\mathcal{A}}^2=$ span$\{ab:a,b\in \mathcal{A}\}$ be a subalgebra of $\mathcal{A}$. In this paper, we prove that if ${\mathcal{A}}^2$ has infinite codimension in $\mathcal{A}$ iff $\mathcal{A}$ has discontinuous square annihilation property (DSAP). In fact, in this case, the algebra $\mathcal{A}$ admits infinitely many non-equivalent algebra norms.