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

TL;DR

This paper characterizes finite-dimensional abelian $C^*$-algebras via orthogonality and proves isometric isomorphism conditions under Birkhoff-James isomorphism, also describing their ideal structure.

Contribution

It establishes that Birkhoff-James isomorphic finite-dimensional abelian $C^*$-algebras are actually isometrically *-isomorphic and details their ideal decomposition.

Findings

Birkhoff-James isomorphism implies *-isomorphism for finite-dimensional abelian $C^*$-algebras.

Decomposition of minimal ideals into skew-fields and non-skew-fields.

Procedure to identify minimal ideals that are fields.

Abstract

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 $\mathcal A_1$ 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.