Isomorphisms of commutative regular algebras

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.