RUS  ENG
Full version
SEMINARS



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

Sh. A. Ayupova, K. K. Kudaybergenovab

a V. I. Romanovskiy Institute of Mathematcs of the Academy of Sciences of Uzbekistan
b Karakalpakstan Regional Branch of the Institute of Mathematics named after V.I. Romanovsky of the Uzbekistan Academy of Sciences

Abstract: 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 $2^c.$

Website: https://us02web.zoom.us/j/8022228888?pwd=b3M4cFJxUHFnZnpuU3kyWW8vNzg0QT09


© Steklov Math. Inst. of RAS, 2024