Isomorphism between the algebra of measurable functions and its subalgebra of approximately differentiable functions
Sh. A. Ayupovab,
Kh. K. Karimovbc,
K. K. Kudaybergenovcdb a National University of Uzbekistan,
4 University St., Tashkent 100174, Uzbekistan
b V. I. Romanovsky Institute of Mathematics,
9 University St., Tashkent 100174, Uzbekistan
c Karakalpak State University,
1 Ch. Abdirov St., Nukus 230112, Uzbekistan
d North Caucasus Center for Mathematical Research VSC RAN,
53 Vatutina St., Vladikavkaz 362025, Russia
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.
Key words:
regular algebra, algebra of measurable functions, isomorphism, band preserving isomorphism.
UDC:
517.982
MSC: 26A33,
34B15,
34D20,
47H10 Received: 25.04.2022
Language: English
DOI:
10.46698/z5485-1251-9649-y