A Class of algebras admitting infinitely many norm topologies

TL;DR

This paper characterizes algebras with infinitely many non-equivalent norms, showing that such algebras have a specific discontinuous property related to their subalgebra of squares.

Contribution

It establishes a precise condition linking the codimension of the square subalgebra to the existence of infinitely many algebra norms.

Findings

Algebras with infinite codimension of $\

ext{square}$ ,

ext{admit infinitely many non-equivalent norms}

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.