Isomorphisms of commutative regular algebras

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis 2022-02, Vol.26 (1), Article 11
• Main Authors: Ayupov, Shavkat, Kudaybergenov, Karimbergen, Karimov, Khakimbek
• Format: Article
• Language: English
• Subjects: Algebra; Boolean; Boolean algebra; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Isomorphism; Mathematics; Mathematics and Statistics; Operator Theory; Potential Theory
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-022-00872-7

The present paper is devoted to study of band preserving isomorphisms of commutative unital regular algebras. Let A be a commutative unital regular algebra over an algebraically closed field F of characteristic zero and let ∇ = ∇ ( A ) be the Boolean algebra of all idempotents in A . Assume that μ is a finite strictly positive countable-additive measure on ∇ and let A be complete with respect to the metric ρ ( x , y ) = μ ( s ( x - y ) ) , x , y ∈ A . We prove that if B is a subalgebra of A such that B ⊃ ∇ , then for any band preserving monomorphism Φ : B → B there exists a band preserving monomorphism Ψ : A → A such that Ψ | B = Φ . Further we introduce a notion of transcendence degree of a commutative unital regular algebra and prove that two homogeneous unital regular subalgebras of S ( Ω ) – the algebra of all classes of measurable complex-valued functions on a Maharam homogeneous measure space ( Ω , Σ , μ ) , are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and their transcendence degrees coincide. As an application we obtain that the regular algebra S (0; 1) – of all classes of measurable complex-valued functions and the algebra AD (0; 1) – of all classes of approximately differentiable functions on [0; 1] are isomorphic.