2024/11/03 by Bojan Kuzma, Sushil Singla, Kuzma, Bojan +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Mathematical Analysis and Transform Methods #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2411.01684
The main goal of the article is to prove that if \mathcal A1 and \mathcal A2 are Birkhoff-James isomorphic C^*-algebras over the fields \mathbb F1 and \mathbb F2, respectively and if \mathcal A1 finite-dimensional, abelian of dimension greater than one, then \mathbb F1=\mathbb F2 and \mathcal A1 and \mathcal A2 are (isometrically) ∗-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⊆ \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.