Classification of abelian finite-dimensional $C^{\ast }$-algebras by orthogonality

Abstract

The main goal of the article is to prove that if ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are Birkhoff-James isomorphic $C^{\ast }$-algebras over the fields ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$, respectively and if ${\mathcal{A}}_1$ finite-dimensional, abelian of dimension greater than one, then ${\mathbb{F}}_1={\mathbb{F}}_2$ and ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are (isometrically) $\ast$-isomorphic $C^{\ast }$-algebras. Furthermore, it is also proved that for a finite-dimensional $C^{\ast }$-algebra $\mathcal{A}$, we have ${\mathcal{L}}_{\mathcal{A}}^{\perp }$ is the sum of minimal ideals which are not skew-fields and ${\mathcal{L}}_{\mathcal{A}}^{\perp \perp }$ is the sum of minimal ideals which are skew-fields, where ${\mathcal{L}}_{\mathcal{A}}$ denotes the set of all left-symmetric elements in $\mathcal{A}$ and for any subset $\mathcal{S}\subseteq \mathcal{A}$, the set ${\mathcal{S}}^{\perp }$ represents the set of all elements of $\mathcal{A}$ which are Birkhoff-James orthogonal to $\mathcal{S}$. A procedure to extract the minimal ideals which are (commutative) fields is also given.