Isomorphism between the Algebra of Measurable Functions and its Subalgebra of Approximately Differentiable Functions

Abstract

The present paper is devoted to study of certain classes of homogeneous regular subalgebras of the algebra of all complex-valued measurable functions on the unit interval. It is known that the transcendence degree of a commutative unital regular algebra is one of the important invariants of such algebras together with Boolean algebra of its idempotents. It is also known that if $(\Omega, \Sigma, \mu)$ is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of $S(\Omega)$ are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and transcendence degrees of these algebras coincide. Let $S(0,1)$ be the algebra of all (classes of equivalence) measurable complex-valued functions and let $AD^{(n)}(0,1)$ ($n\in \mathbb{N}\cup\{\infty\}$) be the algebra of all (classes of equivalence of) almost everywhere $n$-times approximately differentiable functions on $[0,1].$ We prove that $AD^{(n)}(0,1)$ is a regular, integrally closed, $\rho$-closed, $c$-homogeneous subalgebra in $S(0,1)$ for all $n\in \mathbb{N}\cup\{\infty\},$ where $c$ is the continuum. Further we show that the algebras $S(0,1)$ and $AD^{(n)}(0,1)$ are isomorphic for all $n\in \mathbb{N}\cup\{\infty\}.$ As an application of these results we obtain that the dimension of the linear space of all derivations on $S(0,1)$ and the order of the group of all band preserving automorphisms of $S(0,1)$ coincide and are equal to $2^c.$ Finally, we show that the Lie algebra $\operatorname{Der} S(0, 1)$ of all derivations on $S(0,1)$ contains a subalgebra isomorphic to the infinite dimensional Witt algebra.