Isomorphism between the algebra of measurable functions and its subalgebra of approximately differentiable functions

Sh. A. Ayupov ; Kh. K. Karimov ; K. K. Kudaybergenov

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 25-37

Résumé

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.