Classification of abelian finite-dimensional \(C^\)-algebras by orthogonality

• Published in: arXiv.org 2024-11
• Main Authors: Kuzma, Bojan, Singla, Sushil
• Format: Article
• Language: English
• Subjects: Orthogonality; Set theory; Sums
• ISSN: 2331-8422

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.