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

The present talk is devoted to study of certain classes of homogeneous subalgebras of the algebra of all complex-valued measurable functions on the unit interval. 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 applications 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 $S(0,1)$ are equal $2^c.$