Classification of abelian finite-dimensional $C^*$-algebras by orthogonality

The main goal of the article is to prove that if $\mathcal{A}_1$ and $\mathcal{A}_2$ are Birkhoff—James isomorphic $C^*$-algebras over the fields $\mathbb{F}_1$ and $\mathbb{F}_2$, respectively, and if $\mathcal{A}_1$ is 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 $C^*$-algebras. Furthermore, it is also proved that for a finite-dimensional $C^*$-algebra $\mathcal{A}$, we have $\mathcal{L}_{\mathcal{A}}^\bot$ is the sum of minimal ideals which are not skew-fields, and $\mathcal{L}_{\mathcal{A}}^{\bot\bot}$ 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 $\mathcal{S}^\bot$ 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.