ISOMORPHISM BETWEEN THE ALGEBRA OF MEASURABLE FUNCTIONS AND ITS SUBALGEBRA OF APPROXIMATELY DIFFERENTIABLE FUNCTIONS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 2, P. 25-37

УДК 517.982

DOI 10.46698/z5485-1251-9649-y

ISOMORPHISM BETWEEN THE ALGEBRA OF MEASURABLE FUNCTIONS AND ITS SUBALGEBRA OF APPROXIMATELY DIFFERENTIABLE FUNCTIONS*

Sh. A. Ayupov,1'2 Kh. Karimov1'3 and K. K. Kudaybergenov1'3'4

1 V. I. Romanovsky Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan;

2 National University of Uzbekistan, 4 University St., Tashkent 100174, Uzbekistan;

3 Karakalpak State University, 1 Ch. Abdirov St., Nukus 230112, Uzbekistan; 4 North Caucasus Center for Mathematical Research VSC RAN, 53 Vatutina St., Vladikavkaz 362025, Russia E-mail: [email protected], [email protected], [email protected]

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 (Q, £, y) is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of S(Q) 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 £ N U {to}) 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, p-closed, c-homogeneous subalgebra in S(0,1) for all n £ N U {to}, where c is the continuum. Further we show that the algebras S(0,1) and AD(n) (0,1) are isomorphic for all n £ NU {to}. 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 2c. Finally, we show that the Lie algebra Der S(0,1) of all derivations on S(0,1) contains a subalgebra isomorphic to the infinite dimensional Witt algebra.

Keywords: regular algebra, algebra of measurable functions, isomorphism, band preserving isomorphism. AMS Subject Classification: 46L40, 16E50, 16W20.

For citation: Ayupov, Sh. A., Karimov, Kh. and Kudaybergenov, K. K. Isomorphism between the Algebra of Measurable Functions and its Subalgebra of Approximately Differentiable Functions, Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 25-37. DOI: 10.46698/z5485-1251-9649-y.

1. Introduction

In his pioneering papers [1-3] J. von Neumann built the theory on the correspondence between complemented orthomodular lattices and regular rings and proved that if given two *-regular rings with orders n ^ 3 (which means that it contains a ring of matrices of order n),

#The third author was partially supported by the Ministry of Science and Higher Education of the Russian Federation, agreement № 075-02-2023-914.

© 2023 Ayupov, Sh. A., Karimov, Kh. and Kudaybergenov, K. K.

then any lattice isomorphism of their lattices of projections can be uniquely extended as a ring isomorphism of the whole ring. One of important classes of *-regular rings are the *-algebra of operators affiliated with a finite von Neumann algebra. Let M be a finite von Neumann algebra and let S(M) be the *-algebra of all measurable operators affiliated with M. In particular, in the case of type IIi von Neumann algebras the above mentioned result of J. von Neumann means that any ring isomorphism of S(M) completely determined by its action on the lattice of projections.

In [4] (see also [5]) it was proved that for the type II1 von Neumann algebras M and N any ring isomorphism from S(M) onto S(N) is a real algebra isomorphism which is continuous in the locally measure topology. Moreover, there exist a real ^isomorphism ^ : S(M) ^ S(N) and an invertible element a € S(N) such that $(x) = a^(x)a-1 for all x € S(M). Earlier M. Mori [6] characterized lattice isomorphisms between projection lattices P(M) and P(N) of von Neumann algebras M and N, respectively, by means of ring isomorphisms between the algebras of locally measurable operators LS (M) and LS (N) when M and N are von Neumann algebras of type I^ or III.

At the same time, the structure of isomorphisms on regular algebras in the abelian case is completely different from the non commutative one. It is well-known that if a von Neumann algebra M is abelian, then it is *-isomorphic to the algebra L^ (Q) of all (classes of equivalence of) essentially bounded measurable complex-valued functions on a measure space (Q, X, and therefore, S(M) = S(Q) is the algebra of all measurable complex functions on Q. In 2006 A. G. Kusraev [7] proved that S(Q) admits discontinuous in the measure topology automorphisms which identically act on the Boolean algebra V(S(Q)) of all idempotents of S(Q) if and only if V(S(Q)) is non atomic. In [8] we have introduced a new notion of transcendence degree of a commutative unital regular algebra. The transcendence degree of a commutative unital regular algebra is one of the important invariants of such algebras togehter with Boolean algebra of its idempotents. Namely, we have proved that if (Q, X,^) is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of S(Q) are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and their transcendence degrees coincide. In the present paper we shall study some certain classes homogeneous subalgebras of the algebra of all measurable functions on the unit interval.

The paper is organised as follows.

In Section 2 we give some preliminaries from the theory of regular algebras and formulate open problems concerning the structure of homogeneous commutative regular algebras.

In Section 3 we shall consider the algebra AD(n)(0,1) of all classes of complex-valued measurable functions which consists of an almost everywhere n-times approximately differentiable functions on [0,1]. We prove that (0,1) is a regular, integrally closed,

p-closed, c-homogeneous subalgebra in S(0,1) for all n € N U {to}, where c is the continuum (Theorem 3.1). Further we show that the algebras S(0,1) and AD(n)(0,1) are isomorphic for all n € N U {to} (Corollary 3.1).

In Section 4 we prove 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) both equal 2c (see Theorem 4.1 and 4.2).

2. Regular Algebras

In the present section we give some preliminaries from the theory of regular algebras and formulate open problems concerning the structure of homogeneous commutative regular algebras. More detailed info concerning regular algebras can be found in [3, 9, 10].

Recall that an algebra A is said to be regular (in the sense of von Neumann) if, for any a € A there is an element x € A such that axa = a.

Let A be a commutative unital regular algebra with a unity 1 over a field F and let V = V(A) be the set of all idempotents A. On V a partial order is defined as follows: e ^ f if and only if ef = e. Then V becomes a Boolean algebra with the greatest element 1, where 1 — e is the complement of an element e. For a € A, let i(a) be the unique solution of the equations axa = a and xax = x [11, Theorem 1.17]. The element s(a) = ai(a) € V is called the support of a [4, 11].

Let ^ be a finite strictly positive countable-additive measure on V. Define the function p : A x A ^ R, by setting [12], [13, p. 898]:

p(x,y)= y(s(x — y)), x,y € A. (1)

Then A is a topological ring in the metric topology given by p (see [12, Proposition 2.6]).

Below we shall assume that A is a commutative unital regular algebra over an algebraically closed field F of characteristic zero, ^ is a finite strictly positive countable-additive measure on V and A is complete with respect to the metric p defined as in (1).

Let B be a regular subalgebra of A and let B[x] denote the algebra of all polynomials with coefficients from subalgebra B. An element a € A is said to be integral with respect to B, if there exists a unitary polynomial p € B[x], such that p(a) = 0. The integral closure of B is the set of all elements that are integral with respect to B. A subalgebra B is said to be integrally closed, if B(i) = B (see [12]).

Recall that x,y € A differ at e € V provided that from fx = fy, it follows that ef = 0 for all f € V. Clearly, this condition is equivalent to e ^ s(x — y). A subset S of A is said to be locally linearly independent, if for an arbitrary nonzero e € V and each family of elements x1,... ,xn € S that differ pairwise at e and X1,... ,Xn € F, the condition e(\1x1 + ... + A-nxn) = 0 implies that X1 = ... = Xn = 0. For S C A denote by (S} the set of elements of the form x-1 ■ ■ ■ x-fc, where x1,...,xk € S and k,n1,... ,nk € N. A set S C A is called locally algebraically independent if (S} is locally linearly independent. A locally algebraically independent subset S is said to be maximal if it is not a proper subset of a locally algebraically independent set. By the Zorn's Lemma, every locally algebraically independent set is a subset of a maximal locally algebraically independent subset [14].

Let S be a non empty subset in A. Denote by A(S) the least regular, integrally closed, p-closed subalgebra of A generated by S and V. A subset S in A is called faithful, if s(x) = 1 for all x € S. A commutative unital regular algebra A is called homogeneous, if there is a faithful locally algebraically independent subset M such that A(M) = A. Note that the cardinality of any algebraic independent system M in A is not greater than the dimension of the algebra A, that is, |M| ^ dim A. The largest cardinal number 7 such that there exists a faithful maximal locally algebraically independent subset M in A with the cardinality 7 and A(M) = A is called the transcendence degree of A and is denoted as trdeg(A) = 7. In this case we say that A is 7-homogeneous. In other words, A is 7-homogeneous, if there is a faithful locally algebraically independent subset M with the cardinality 7 such that A(M) = A and any faithful system in A of cardinality bigger than 7 is algebraically dependent.

Recall that a weight of the Boolean algebra V is the least cardinality of sets which generate V, i.e.,

t(V) = inf {|X| : X generates V} .

For 0 = e € V set Ve = {x € V : x ^ e}. A Boolean algebra V is said to be homogeneous, if t(Ve) = t(V) for any non zero e € V [13, 15, 16].

Let (Q, X, be a measure space with a finite strictly positive countable-additive measure ^ and let S (Q) = S (Q, X,^) be the algebra of all (classes of) F-valued measurable functions on (Q, X,^). It is clear that S(Q) is a commutative unital regular algebra. A rank metric p on S(Q) defined as in (1):

p(x, y) = ^ ({w € Q : x(w) = y(w)}), x, y € S(Q).

Recall that a measure space (Q, X, is called Maharam homogeneous if the Boolean algebra V of the algebra S(Q) is homogeneous [13, Page 908]. An automorphism 0 : V ^ V is called measure-preserving, if ^(e) = ^(0(e)) for all e € V. Every measure-preserving automorphism of V can be extended to an automorphism of S(Q), which we also denote by 0 [13, p. 907].

Let G = {0} be a group of automorphisms of V. The group G is called ergodic, if for every 0 = e € V the following equality holds: V0(e) = 1.

In [8, Proposition 4.4] the following criteria of homogeneity of regular subalgebras was obtained.

Proposition 2.1. Let (Q, X,^) be a Maharam homogeneous measure space with a finite strictly positive countable-additive measure ^ and let A be an integrally closed and p-clo-sed regular subalgebra in S(Q). Suppose that G is a measure-preserving ergodic group of automorpisms on V such that 0(A) = A for all 0 € G. Then A is homogeneous.

From the above it follows that if (Q, X,^) is a Maharam homogeneous measure space, then S(Q) is a homogeneous commutative regular algebra. In particular, S(0, 1) is a homogeneous commutative regular algebra (see [8]).

At the end of this Section we formulate open problems concerning homogeneous commutative regular algebras.

Problem 2.1: Is any commutative regular algebra A a direct sum of homogeneous commutative regular algebras?

Problem 2.2: Does cardinal number of any maximal faithful locally algebraically independent subset of a homogeneous commutative regular algebra A coincide with trdeg(A)?

Problem 2.3: Let (Q, X,^) be a Maharam homogeneous measure space with a finite strictly positive countable-additive measure Find a connection between the transcendence degree of the algebra S(Q) and the weight of the Boolean algebra V(S(Q)).

In the next Section we shall give a solution of the last problem in the case of separable Maharam homogeneous measure space.

3. Algebra of Approximately Differentiable Functions

Let E c R be a Lebesgue measurable set, f : E ^ C be a measurable function. Let t0 € R be a point in which E has density 1. Recall that a number I is called the approximate limit of f at t0 if the set {t € E : |f (t) —1| < e} has density one at t0 for each e > 0, and I is denoted by ap — limt^t0 f (t). If the approximate limit

, , fit) - №) /ap(io) :=ap-hm f _ ^

exists and it is finite, then it is called approximate derivative of the function f at to and the function f is called approximately differentiable at to [17]. Further, for the sake of convenience the approximate derivative fp we denote as f.

Let AD(0,1) be the set of all classes of complex-valued measurable functions which consists of an almost everywhere approximately differentiable functions on [0,1].

Note that by [18, Proposition 4.7 and Corollary 4.11] the algebra AD(0,1) is a regular, integrally closed and p-closed proper subalgebra in S(0,1). Moreover,

V (S(0,1)) = V (AD(0,1)). (2)

Further, the algebra AD(0,1) is homogeneous, and the algebras S(0,1) and AD(0,1) are isomorphic [8, Theorem 4.8].

First we shall give a solution of Problem 2.3 in the case of separable Maharam homogeneous measure space. In particular, we obtain the following connection between the transcendence degree of the algebra S(0,1) and the weight of the Boolean algebra V(S(0,1)) :

trdeg S(0,1) = c = 2H° = 2T (V(S(0'1))).

Proposition 3.1. The transcendence degrees of the algebra of complex-valued measurable functions and the algebra of approximately differentiable functions on the unit interval coincide and are equal to continuum, i. e.

trdeg S (0,1) = trdeg AD(0,1) = c,

where c is the continuum.

< Let [di : i € I} be a Hamel basis for the field of real numbers over the field of rational numbers. For each i € I define a function xi € S(0,1) as follows

x.

(t) = tldil, t € [0,1]. (3)

Then [xi : i € I} is a faithful locally algebraically independent subset of S(0,1) of continuum cardinality [19]. Since [xi : i € I} C AD(0,1), it follows that trdeg AD(0,1) ^ c.

Note that each x € C[0,1] is uniquely determined by its restriction on Q n [0,1]. Hence,

|C[0,1]| < CQnM = cH° = (2H°)Ho = 2H°H° = 2H° = c. (4)

Note that by Lusin's theorem, it follows that

S(0,1)= C [0,1f. (5)

Further, from (4) and (5), we can conclude that

|S (0,1)| = c. (6)

Indeed, let us embed the space S(0,1) into the space C[0,1]H° as follows: for any x € S(0,1) take a sequence [x„} C C[0,1] such that xn ') x and consider the mapping

x € S(0,1) {xn} € C[0,1]

Hq

Note that we identify C[0,1] with classes of measurable functions that contain continuous functions. It is clear that this mapping is injective. Therefore

c < |S(0,1)| < C[0, 1]>

= cHo = c.

0

So,

c < trdeg AD(0,1) < trdeg S(0,1) < |S(0,1)| < c,

and therefore trdeg S(0,1) = trdeg AD(0,1) = c. The proof is completed. >

Remark 3.1. In the previous Proposition we have show that S(0,1) is c-homogeneous. In the proof of Proposition 3.1 we have constructed continual faithful algebraically independent system in S(0,1), and by Zorn's Lemma this system can be completed to a maximal faithful algebraically independent system. But in general case, to point a concrete maximal continual faithful maximal algebraically independent system in S(0, 1) is a non trivial problem.

Let AD(n)(0,1) be the set of all classes of complex-valued measurable functions which consists of an almost everywhere n-times approximately differentiable functions on [0,1] and let

(0,1) = p| AD(n)(0,1).

Theorem 3.1. The algebra AD(n)(0,1) is a regular, integrally closed, p-closed and c-ho-mogeneous subalgebra in S(0,1) for all n € N U {to}. < First we shall prove for natural n.

The proof is by induction on n. For n = 1 all properties except homogeneity have shown in [8], and c-homogeneity already proved in Proposition 3.1. Assume that we have proved for n — 1.

Let us first show that

AD(n)(0,1) is regular. Let f be a almost everywhere n-times approximately differentiable function on [0,1], that is, [f] € AD(0,1). Then the function g on [0,1] defined as

m i/fe. «/«>¿0, \0, if f(t)=0

is also n-times approximately differentiable almost everywhere in [0,1]. Hence, [g] € AD(0,1) and f2g = f. This means that AD(0,1) is regular.

Let x € S(0,1) be an integral element with respect AD(n)(0,1). By the induction assumption

AD(n-1) (0,1) is integrally closed and therefore x € AD(n-1)(0,1). By [12, Proposition 3.3], there exist a partition {e1 ,...,ek} of s(x) and natural numbers n1 < n2 < ... < nk such that for every i = 1..., k the minimal degree of a unitary polynomial gj over AD(n-1)(0,1) for which gj(ejx) =0 is nj. Let i € {1,... ,k} be a fixed index and let

(ejx)ni + arai-1(ej x)ni-1 + ... + a^x) + ao = 0,

where a0,...,ani-1 € ejAD(n)(0,1). Let i € {1,...,k} be a fixed index. Taking the approximate derivative in the last equality we obtain that

xp + q = 0,

where

n i — 1 ni-1

p = nj(ejx)ni—1 + ^ ajj(ejx)j—1 and q = ^ a^(ejx)j.

j=1 j=o

Note that p,q € AD(n—1)(0,1), and therefore ejx = — qi(p) belongs to AD(n—1)(0,1). Thus ejx € AD(n)(0,1) for all i = 1,... ,k, hence x € AD(n)(0,1).

Let us show that AD(n)(0,1) is p-closed. Take {xk} C AD(n)(0,1) such that xk —% x € S(0,1). By the induction assumption

AD(n—1)(0,1) is p-closed and therefore x €

AD(n-1) (0,1). By [18, Proposition 4.12], the approximate differentation operator dad : AD(0,1) ^ S(0,1) is non-expansive, that is,

p(9ad(a), bad(b)) ^ p(a, b) (7)

for all a, b € AD(0,1). Using (7) we have that &ad(xk) —^ 9ad(x). Since AD(n-1)(0,1) is p-closed and [dad(xk)} C AD(n-1)(0,1), it follows that dad(x) € AD(n-1)(0,1). Thus x € AD(n)(0,1), and therefore AD(n)(0,1) is p-closed.

Since AD(n)(0,1) is invariant under rational translation modulo 1, by Proposition 2.1, it follows that it is homogeneous.

Let [xi : i € I} be a locally algebraically independent subset in S(0,1) as in (3). Since [ xi : i € I} is a subset of AD(n)(0,1), it follows that

c < trdeg AD(n)(0,1) < trdeg S(0,1) = c,

and therefore trdeg AD(n)(0,1) = c.

Now we shall consider the case n = to.

Let x € AD^ (0,1) be a non zero element. Then its partial inverse i(x) € AD(n)(0,1) for all n € N. Thus i(x) € AD^(0,1) and i(x)x2 = x. Hence AD^ (0,1) is regular. Further, since AD(n)(0,1) is p-closed for all n, it follows that AD^ (0,1) is also p-closed.

Let x € S(0,1) be an integral element with respect to AD^ (0,1). Then x is an integral element with respect to AD(n)(0,1) for all n ^ 1. Since AD(n)(0,1) is integrally closed, it follows that x € AD(n)(0,1) for all n ^ 1. Thus x € AD~ (0,1).

Again using invariancy of AD^ (0,1) under rational translation modulo 1, we obtain that it is homogeneous. Further, let xi(t) = t|di| as in (3). Since [xi}ie/ C AD^(0,1), it follows that trdeg AD^(0,1) = c. The proof is completed. >

Corollary 3.1. The algebras S(0,1) and AD(n)(0,1) are isomorphic for all n € N U [to}.

< By (2), the Boolean algebras V(S(0,1)) and V(AD(0,1)) coincide. Since e = 0 for all e € V(S(0,1)), it follows that V(S(0,1)) = V(AD(n)(0,1)) for all n € N U [to}. By Theorem 3.1, we have that trdeg S(0,1) = trdeg AD(n)(0,1) = c for all n € NU[to}. Therefore by [8, Theorem 4.6], we obtain that the algebras S(0,1) and AD(n)(0,1) are isomorphic. The proof is completed. >

We have mentioned that by Lusin's theorem, it follows that S(0,1) = C[0,1] (see (5)). On the other hand, by Whitney's theorem [20, Theorem 1] (see also [17, Theorem 3.1.16]), any approximately differentiable function on [0; 1] is continuously differentiable outside of a closed subset of arbitrarily small measure, it follows that

AD{ 0,1) = C(1)[0,1]P.

At the same time for n ^ 2 we have the following result. Proposition 3.2. Let n ^ 2. Then

C(n)[0,1]P C AD(n)(0,1),

where C (n) [0, 1] is the algebra of all n-times continuously differentiable complex-valued functions on [0,1].

< By [21] for any e € (0,1) there are a closed subset A C [0,1] and a monotone non negative function f € C(1) [0,1] such that

(a) m(A) = 1 — e;

(b) f|a = 0;

(c) {t € A : f (t) = g(t)} is a finite set for any g € C(2)[0,1].

Consider the function h = xaf. Then h = 0, hence h € AD(n)[0,1] for all n ^ 1. From properties (a) and (c) for any g € C(n)[0,1] (n ^ 2) we have that

p(h,g) = p ({t € [0,1]: h(t) = g(t)}) ^ p ({t € A : f (t) = g(t)}) = 1 — e.

Thus h £ C(ra)[0,for all n ^ 2. The proof is completed. >

Similar to the proof of Theorem 3.1 we obtain the following result.

Theorem 3.2. The algebra C(ra)[0,1]P is a regular, integrally closed, p-closed and c-homogeneous subalgebra in S(0,1) for all n € N U {to}.

4. The Space of All Derivations and Group of Band Preserving Automorphisms of Algebras of Measurable Functions

4.1. The dimension of the space of all derivations on algebras of measurable functions. Recall that a linear mapping D : A % A is called a derivation if it satisfies the identity D(xy) = D(x)y + xD(y) for all x, y € A.

Let : AD(0,1) % S(0,1) be the approximate differentation operator. It is clear that maps AD^(0,1) into itself. Let a : S(0,1) % AD^(0,1) be an isomorphism (see Corollary 3.1). Setting

we obtain a non zero derivation on S(0,1).

Recall that the existence of non zero derivations on S(0,1) were obtained independently by A. F. Ber, F. A. Sukochev, V. I. Chilin [12] and A. G. Kusraev [7]. In [19, Proposition 6] it was shown that the dimension of the linear space of all derivations on the algebra S(0, 1) is uncountable.

For the proof of the next result we need the following Erdosh-Kaplansky Theorem [22, p. 258]. Let V be an infinite dimensional linear space over a field F and let V* be its dual space. Then

Proposition 3.1 allows us to determine the dimension of all derivations on S(0,1). Namely, we shall prove the following result.

Теорема 4.1. Let Der S(0,1) be the linear space of all derivations of S(0,1). Then

dimDer S (0,1) = 2c.

< Let M be a maximal locally algebraically independent subset in S(0,1). Denote by F(M, S(0,1)) the set of all mappings f : M ^ S(0,1) such that

dAD(x) = a-1 (dAD(a(x)), x € S(0,1),

dim V * = |F|

dim V

(8)

s(f (x)) < s(x)

(9)

for all x € M. Then the mapping

S € Der S(0,1) ^ 5\m € F(M, S(0,1))

defines a linear isomorphism between Der S(0; 1) and F(M, S(0,1)) (see [19, Theorem 1]).

By Proposition 3.1 the algebra S(0,1) is c-homogeneous and therefore we can choose a faithful maximal locally algebraically independent subset M of S(0,1) of the continuum cardinality. In this case, since s(x) = 1 for all x € M, it follows that the linear space F(M, S(0,1)) coincide with the linear space S(0,1)M of all mappings form M into S(0,1). Further, using (6) we obtain that

dimDerS(0,1) = dimS(0,1)M < |S(0,1)|trdegS(0>1) = cc = (2H°)c = 2H°-c = 2c.

It is clear that the mapping

f € (spanM)* —^ f € S(0,1)M

defined as follows

f(x) = f (x) 1, x € M is injective. Therefore dim S(0,1)M ^ dim (spanM)*. Finally,

dimDerS(0,1) = dimS(0,1)M ^ dim(spanM)* = |C||M 1 = cc = 2c.

The proof is completed. >

Example 1. It is well-known that for an arbitrary algebra A the linear space Der(A) equipped with the following Lie bracket

[D1, D2] = D1D2 — D2D1, D1, D2 € Der(A)

becomes a Lie algebra.

We shall show that the Lie algebra Der S(0,1) contains a subalgebra isomorphic to the infinite dimensional Witt algebra.

Let M be a faithful maximal locally algebraically independent subset of S(0, 1) and let £ € M be a fixed element. For any n € Z set

£n+1, if Z = £,

¿n(Z) = I 0, if Z = £.

Since s(5n(Z)) < s(Z) for all Z € M, it follows that ¿n € F(M, S(0,1)) for all n € Z (see (9)). By (10), for each n € Z there is a unique derivation dn € DerS(0,1) such that dn|m = ¿n. Since dn is a derivation, using Leibniz rule, we obtain that dn(£k) = k£n+k for all n,k € Z. Taking into account the latter observation, we have that

[dn, dm] (£) =dn (dm(£)) — dm (dn(£)) = dn (£m+1) — dm (£n+1)

= (m + 1)£n+m+1 — (n + 1)£n+m+1 = (m — n)£n+m+1 = (m — n)dn+m(£)

and

[dn,dm](Z) = 0, Z = £

for all n, m € Z. Thus [dn, dm]|M = (m — n)dn+m|M, and hence according to (10),

[dn, dm] = (m — n)dn+m (11)

for all n,m € Z. The relation (11) shows that the subspace W = span[dn : n € Z} is a Lie subalgebra of Der S(0,1) which is isomorphic to the infinite dimensional Witt algebra.

It should be noted that Witt algebras compose one of the four classes of Cartan type Lie algebras originally introduced in 1909 by E. Cartan [23] when he studied infinite dimensional simple Lie algebras over complex numbers.

Based on latter note we may formulate the following problem.

Problem 4.1: Any finite or countable dimensional complex simple Lie algebra can be embedded into the Lie algebra Der S(0, 1).

Remark 4.1. Let us consider on S(0,1) a metric pM defined as [24]

pM(x,y) =inf{e> 0: p ({t € [0,1] : |x(t) — y(t)| > e}) < e}.

Note that pM-convergence in S(0,1) coincide with the convergence in measure. Since the subspace of all simple functions F(V) is pM-dense in S(0,1) and any derivation of S(0,1) vanishes on F(V), it follows that any pM-continuous derivation on S(0,1) is identically zero. So, any non zero derivation of S(0,1) is pM-discontinuous.

Note that this phenomenon is a purely commutative effect. In the case of type II1 von Neumann algebra M all derivations of the regular algebra S(M) are inner, in particular, is continuous in the measure topology [25, 26].

4.2. The order of the group of all band preserving automorphisms of algebras of measurable functions. Recall that two elements x, y € A are said to be orthogonal (notation x ± y), if s(x)s(y) = 0. A linear mapping $ : A % A is said to be band preserving, if [7, 27]

x ± y $(x) ± y.

Since x ± 1 — s(x), we see that $ is band preserving if and only if s ($(x)) ^ s(x) for all x € A.

Let $ : A % A be an automorphism. Then $ is band preserving if and only if $ acts identically on Fc(V) [8, Proposition 3.3].

Assume that A is homogeneous and M is a faithful maximal locally algebraically independent subset of A. Denote by G (M, A) the set of all mappings g : M % A such that g(M) = {g(x) : x € M} is also a faithful maximal locally algebraically independent subset of A.

A permutation of Jl is a bijection from Jl onto Jl. The group of all permutations of Jl is denoted by Sym{y^). Let FC(V) = spanVp. Then FC(V) is a regular subalgebra in A (see [12]). Let Fc(V)^ be the group of all invertible elements from Fc(V).

Denote by Autv (A) the group of all band preserving automorphisms of A. We may view to the group Autv (A) as a Galois group of the ring extension A over Fc(V). There is an injective mapping

g € G(M, A) % $g € Autv(A),

in particular, the group Autv(A) contains a subgroup isomorphic to the group Sym(M) x (FC(V)^)|M 1 (see [8, Theorem 4.2]).

Similarly, as in the proof of Theorem 4.1 we have that

|AutvS(0,1)| < S(0,1)S(0>1)

= cc = 2c.

On other hand, taking into that Autv S(0,1) contains a subgroup isomorphic to (Fc(V)^)| we obtain that

|Autv S(0,1)| ^ |FC(Vn|M 1 ^ c|M 1 = cc = 2c.

So, we have proved the following result.

Theorem 4.2. The order of the group of all band-preserving automorphisms of the algebra of all complex-valued measurable functions on the unit interval is equal 2c, i. e.

|Autv S (0,1)| = 2c.

Remark 4.2. We note that A. E. Gutman, A. G. Kusraev, S. S. Kutateladze in [14, Comment 5.3.7], concerning the notion of algebraic independence in regular rings wrote: This notion, presenting the external interpretation of the internal notion of algebraic independence (or transcendence), seems to turn out useful in studying the descents of fields or general regular rings. As we can see, our results confirm the above prediction.

Acknowledgments. We are indebted to the Referee for very valuable suggestions and remarks, which helped us to significantly improve the exposition.

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Received April 25, 2022 Shavkat A. Ayupov

V. I. Romanovsky Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan, Director of V. I. Romanovsky Institute of Mathematics; National University of Uzbekistan named after M. Uluhgbek, 4 University St., Tashkent 100174, Uzbekistan E-mail: shavkat. ayupov@mathinst .uz https://orcid.org/0000-0002-4006-1460

Khakimbek K. Karimov V. I. Romanovsky Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan, Postgraduate Student;

Karakalpak State University named after Berdakh, 1 Ch. Abdirov St., Nukus 230112, Uzbekistan, Assistant

E-mail: [email protected]

https://orcid.org/0009-0006-1908-3717

Karimbergen K. Kudaybergenov V. I. Romanovsky Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan, Chief Researcher;

North Caucasus Center for Mathematical Research VSC RAN,

53 Vatutina St., Vladikavkaz 362025, Russia,

Chief Researcher

E-mail: [email protected]

https://orcid.org/0000-0003-0311-9683

Владикавказский математический журнал 2023, Том 25, Выпуск 2, С. 25-37

ИЗОМОРФИЗМ МЕЖДУ АЛГЕБРОЙ ИЗМЕРИМЫХ ФУНКЦИЙ И ЕЕ ПОДАЛГЕБРОЙ АСИМПТОТИЧЕСКИ ДИФФЕРЕНЦИРУЕМЫХ ФУНКЦИЙ

Аюпов Ш. А.1,2, Каримов Х. К.1,3, Кудайбергенов К. К.1,3,4

1 Институт математики имени В. И. Романовского, Узбекистан, 100174, Ташкент, ул. Университетская, 9;

2 Национальный университет Узбекистана им. М. Улугбека, Узбекистан, 100174, Ташкент, ул. Университетская, 4;

3 Каракалпакский государственный университет им. Бердаха, Узбекистан, 230112, Нукус, ул. Ч. Абдирова, 1;

4 Северо-Кавказский центр математических исследований ВНЦ РАН, Россия, 362025, Владикавка, ул. ватутина, 53 E-mail: [email protected], [email protected], [email protected]

Аннотация. Настоящая работа посвящена изучению некоторых классов однородных регулярных подалгебр алгебры всех комплекснозначных измеримых функций на единичном интервале. Известно, что степень трансцендентности унитальной коммутативной регулярной алгебры является одним из важных инвариантов таких алгебр наряду с булевой алгеброй ее идемпотентов. Также известно, что если (О, — однородное пространство с мерой Магарам, то две однородные унитальные регулярные подалгебры в S(О) изоморфны тогда и только тогда, когда их булевы алгебры идемпотентов изоморфны, и степени трансцендентности этих алгебр совпадают. Пусть S(0,1) — алгебра всех (классов эквивалентности) измеримых комплекснозначных функций, и пусть (0,1) (n £ N U {то}) — алгебра всех (классов эквивалентности) почти всюду n-раз асимптотически дифференцируемых функции на [0,1]. В работе доказано, что AD(n) (0,1) является регулярной, цело-замкнутой, р-замкнутой, c-однородной подалгеброй в S(0,1) для всех n £ N U {то}, где c — континуум. Далее мы покажем, что алгебры S(0,1) и

(0,1) изоморфны для всех n £ N U {то}. В качестве приложения этих результатов установлено, что размерность линейного пространства всех дифференцирований на S(0, 1) и порядок группы всех сохраняющих полосу автоморфизмов алгебры S(0,1) совпадают и равны 2c. Мы также покажем, что алгебра Ли Der S(0,1), всех дифференцирований алгебры S(0,1), содержит подалгебру, изоморфную бесконечномерной алгебре Витта.

Ключевые слова: регулярная алгебра, алгебра измеримых функций, изоморфизм, сохраняющие полосы изоморфизм.

AMS Subject Classification: 26A33, 34B15, 34D20, 47H10.

Образец цитирования: Ayupov, Sh. A., Karimov, Kh. and Kudaybergenov, K. K. Isomorphism Between the Algebra of Measurable Functions and Its Subalgebra of Approximately Differentiable Functions // Владикавк. мат. журн.—2022.—Т. 25, № 2.—C. 25-37 (in English). DOI: 10.46698/z5485-1251-9649-y.