A Class of algebras admitting infinitely many norm topologies
J. G. Patel
TL;DR
Addresses the problem of when a normed algebra $\mathcal{A}$ has infinitely many non-equivalent algebra norms by linking this to the codimension of $\mathcal{A}^2$ and the DSAP. The main result proves that $\mathcal{A}^2$ having infinite codimension in $\mathcal{A}$ is equivalent to $\mathcal{A}$ possessing DSAP, and that in this regime $\mathcal{A}$ supports infinitely many inequivalent algebra norms. The construction uses discontinuous functionals to build a family of algebra norms $p_n(a)=\|a\|+|\varphi_n(a)|$, yielding pairwise non-equivalent norms. The paper provides diverse examples, such as $\ell^2$ with pointwise multiplication and the disc algebra $A(\mathbb{D})$, illustrating the broad applicability of the criterion.
Abstract
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.