2-local derivations on algebras of matrix-valued functions on a compactum Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2018, Том 20, Выпуск 1, С. 38-49

YffK 517.98

2-LOCAL DERIVATIONS ON ALGEBRAS OF MATRIX-VALUED FUNCTIONS ON A COMPACTUM

Sh. A. Ayupov, F. N. Arzikulov

This paper is dedicated to the memory of Professor Inomjon Gulomjonovich Ganiev

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-local derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and 2-local derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every 2-local derivation on a *-algebra C(Q,M„(F)) or C(Q, N(F)), where Q is a compactum, M„(F) is the ^-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the *-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

DOI: 10.23671/VNC. 2018.1.11396.

Mathematical Subject Classification (2010): 46L57, 46L40.

Key words: derivation, 2-local derivation, associative algebra, C*-algebra, von Neumann algebra.

Introduction

The present paper is devoted to 2-local derivations on algebras. Recall that a 2-local derivation is defined as follows: given an algebra A, a map A : A ^ A (not linear in general) is called a 2-local derivation if for every x, y G A, there exists a derivation Dx,y : A ^ A such that A(x) = Dx,y(x) and A(y) = Dx,y(y).

In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. A similar description for the finite-dimensional case appeared

© 2018 Ayupov Sh. A., Arzikulov F. N.

later in [7]. In the paper [8] 2-local derivations have been described on matrix algebras over finite-dimensional division rings.

In [5] the authors suggested a new technique and have generalized the above mentioned results of [10] and [7] for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B (H) of all linear bounded operators on an arbitrary (no separability is assumed) Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. After it is also published a number of papers devoted to 2-local derivations on associative algebras.

In the present paper we also suggest another technique and apply to various associative algebras of infinite dimensional matrix-valued functions on a compactum. As a result we will have that every 2-local derivation on such an algebra is a derivation. As the main result of the paper it is established that every 2-local derivation on a *-algebra C(Q,Mn(F)) or C(Q, Nn(F)), where Q is a compactum, Mn(F) is the *-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in Section 1, Nn(F) is the *-subalgebra of Mn(F) defined in Section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

We conclude that there are a number of various associative algebras of infinite dimensional matrix-valued functions on a compactum every 2-local derivation of which is a derivation. The main results of this paper are new. The method of proving of these results presented in this paper is universal and can be applied to associative, Lie and Jordan algebras. Its respective modification allows to prove similar problem for Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum. In this sense our method is useful.

1. Preliminaries

M

Definition. A linear map D : M ^ M is called a derivation, if D(xy) = D(x)y + xD(y) for every two elements x,y £ M.

A map A : M ^ M is called a 2-local derivation, if for every two elements x,y £ M there exists a derivation Dx,y : M ^ M such that A(x) = Dx,y(x), A(y) = Dx,y(y).

DM that is there exists an element a £ M such that

D(x) = ax — xa, x £ M.

M

one: A map A : M ^ M is called a 2-local derivation, if for every two elements x,y £ M there exists an element a £ M such that A(x) = ax — xa A(y) = ay — ya.

Let throughout the paper n be an arbitrary infinite cardinal number, 5 be a set of indices of the cardinality n. Let {ej} be a set of matrix units s ueh that ej is an x n-dimensional matrix, i. e. ej = (aa/3)apeE, the (i,j)-th component of which is 1, i. e. aj = 1, and the rest components are zeros.

Let {m^} be a set of n x n-dimensional matrixes and m£ = (m^3)aseE for every Then by m£ we denote the matrix whose components are sums of the corresponding components of matrixes of the set {m£}, i. e.

J2m£ = ( Em£

' a/SeE

a3

Here, the maximal quantity of nonzero summands of the sum mOa3 is countable.

Let throughout the paper F is the field of complex numbers C (real numbers R or quaternion body H) and

Mn(F) = E X%jeij : : X%j £ F)(3 K £ R)

^jes

(Vn e N)V{ekl}nkl=1 C {eij}

E Aklekl

kl=\

^ K

where || ^klekl|| the norm of the matrix Y^fki=1 ^kleui in the finite dimensional

C*-algebra, generated by {ekl}nl=v ^ easy to see that Mn(F) is a vector space over F. In Mn(F) we introduce an associative multiplication as follows: if

x =

eij, y = E

■ij

i,jeE

are elements of Mn (F) then

xy

£

i,jes

ij

EAlf ^

Lies

With respect to this operation Mn(F) becomes an associative algebra and Mn(F) = B(l2(2)), where l2(2) is a Hilbert space over F with elements {xi}les, xi e F for all i e 2, B(l2(2)) is the associative algebra of all bounded linear operators on the Hilbert space l2(2). Then Mn(F) is a von Neumann algebra of infinite n x n-dimensional matrices over F if F = C (see [2]) and Mn(F) is a real von Neumann algebra if F = R or H.

Recall that a Hilbert space H is an infinite dimensional inner product space which is a complete metric space with respect to the metric generated by the inner product fl, Section 1.5].

Similarly, if we take the algebra B(H) of all bounded linear operators on an arbitrary Hilbert space H and if {ql} is an arbitrary maximal orthogonal set of minimal projections of B(H), then B(H) = qiB(H)qj (see [4]).

Let throughout the paper X be a hyperstonean compactum, C(X) denote the algebra of all F-valued continuous functions on X and

M = < E Alj (x)eij : (Vi,j : \lj (x) e C(X)) (3 K e

I i,jes

E Xkl(x)eki

^ K

(V m £ N {y{eki }m= i })

kl=1,..,m

where || ^kl=1 m \kl(x)ekl || is the norm of J2kl=1 m ^kl(x)ekl in the C*-algebra C(X,Mn(C) ), where Mn(C) the finite dimensional C*-algebra, generated by It

is clear that M„(C) = M„(C) and

C(X, Mn(C)) = C(X) <g> M„(C).

The set M is a vector space with point-wise algebraic operations. The map || ■ || : M ^ R+ defined as

a|| =

sup

{ekl}ki=i^{eij }

E Akl(x)ekl

kl=l

is a norm on the vector space M, where a G M and a = ^i jes (x)eij-In M we introduce an associative multiplication as follows: if

are elements of M then

i,j€E

xy = X)

¿,jes

i,jeE

G M.

With respect to this multiplication M becomes an associative algebra and M is a real or complex von Neumann algebra of type In by Theorem 5 in [4].

Let M te a C*-algebra, A : M ^ M be a 2-local derivation. Now let us show that A is homogeneous. Indeed, for each x G M, and for A G C there exists a derivation Dx,ax such that A(x) = Dx,^x(x) and A(Ax) = Dx^x(Ax). Then

A(Ax) = Dx,Ax(Ax) = ADx>Ax(x) = AA(x).

Hence, A is homogeneous. At the same time, for each x G M, there exists a derivation Dx,x2 such that A(x) = Dx,x2(x) and A(x2) = Dx,x2(x2). Then

A(x2) = Dx,x2 (x2) = Dx,x2 (x)x + xDxx2 (x) = A(x)x + xA(x).

In [6] it is proved that every Jordan derivation on a semi-prime algebra is a derivation. Since M is semi-prime (i. e. aMa = {0} implies that a = {0}), the map A is a derivation if it is additive. Therefore, to prove that the 2-local derivation A : M ^ M is a derivation it is sufficient to prove that A : M ^ M is additive in the proof of Theorem 1.

x

2. 2-local derivations on some associative algebras of matrix-valued functions

Let Q be a compactum. Then the algebra C(Q) of all continuous complex number-valued functions on Q is a C*-algebra and by Theorem 1.17.2 in [9] the second dual space C(Q)** is a commutative von Neumann algebra. Hence there exists a hyperstonean compactum X such that C(Q)** = C(X). If we take the *-algebra C(Q,Mn(C)) of all continuous maps of Q to Mn(C), then we may assume that C(Q,Mn(C)) C M. In this rase the set {eij} of constant functions belongs to C(Q, Mn(C)) and the weak closure of C(Q,Mn(C)) in M coincides with M. Hence by separately weakly continuity of multiplication every derivation of C(Q,Mn(C)) has a unique extension to a derivation on M [9, Lemma 4.1.4]. Therefore, if A is a 2-local derivation on C(Q, Mn(C)), then for every two elements x, y G C(Q,Mn(C)) there exists a derivation Dx,y : M ^ M such that A(x) = Dx,y(x), A(y) = Dx,y(y), i. e. Dx,y is a derivation of M (not only of C(Q, Mn(C))). The following theorem is the key result of this section.

Theorem 1. Let A be a 2-local derivation on C(Q, Mn(C)). Then A is a derivation.

We first prove some lemmas necessary for the proof of Theorem 1.

By the above arguments for every 2-local derivation A on C(Q,Mn(F)) and for each x G C (Q, Mn(F)) there exist a G M such that

A(x) = ax — xa.

Put

eij := ^^ e¿n,

where for all n if £ = i, n = j then A^n = 1, otherwise A^n = 0 and 1 is unit of the algebra C(Q). Let {a(ij)} C M be a subset such that

A(ej) = a(ij)ej - e^-a(ij).

for all i, j, let aij e^j , aij G C (Q) be the (i, j)-th component of the element e^ a(ji)ejj of M for all pairs of different indices i, j and let e^n be the matrix with all such components,

the diagonal components of which are zeros.

ij

A(e»j) = afne^neij - eij ^ afne^ + a(ij)« ej - e^-a(ijj, (1)

where a(ij)n, a(ij )jj are functions in C (Q) wiicii are the coefficients of the Peirce components eii a

(ij)

eM, ejj a(ij)

< Let k be an arbitrary index different from i, j and let a(ij, ik) G M be an element such that

A(eifc) = a(ij, ik)e»fc - e^a(ij, ik) and A(ej) = a(ij, ik)ej - e^-a(ij, ik).

Then

efcfcA(eij)ejj = efcfc (a(ij,ik)ej - eija(ij,ik))ejj = efcfca(ij, ik)e»j - 0 = efcfc a(ik)

eij efcfceij / y a e£nejj = ekfcaki eij efcfc eij / y a e£n ejj — efcfc ^ ^ afneij ekk eij ^ ^ a«n e^n ejj — efcfc ( a^ne£neij eij ^ ^ a^ne£n I ejj •

Similarly,

efcfcA(eij)eii — efcfc ( ^e^neij j ^ fl^ne^n I &

V ¿=n ¿=n /

(a

Let a(ij, kj) G M be an element such that

A(efcj) = a(ij, kj)efcj - efcja(ij, kj) and A(ej) = a(ij, kj)ej - eija(ij, kj). Then

eiiA(eij)efcfc = eii(a(ij, kj)eij - eija(ij, kj)) efcfc = 0 - eija(ij, kj)efcfc =0 - eija(kj)efcfc =0 - eijajfcefcfc

— eii> y ^ e^n eij efcfc eij ^ ^ a^n e?n efcfc — eii j fl^n e?n eij eij ^ ^ fl^n e?n I efcfc.

¿=n ¿=n V ¿=n ¿=n /

Also similarly we have

ejj A(eij )efcfc = ejj 1 ^ e£n eij ej) ^ a?n e^n I efcfc;

V ¿=n ¿=n /

eii A(eij )eii — eii ( C^n Cij j ^ Q^n 11

\ %=n %=n '

ejj A(eij )ejj — ejj\ ^^ e%neij — o%Ve%n )'

V %=n %=n /

jj

- eij o ■%=n %=n

Hence the equality (1) is valid. >

We take elements of the sets {{ei%}%}i and {{e%j }%}j in pairs ({ea%}%, {e%/}%) such that a — fl. Then using the set {({ea%}%, {e%/s}%)} of such pairs we get the set {ea/}.

Let xo — {eas} be a set {vijeij}ij such that for all i, j if (a, fl) — (i, j) then Vj — 0 £ C(Q) else Vj — 1 £ C(Q). Then xo £ C(Q, Mn(C)). Fix different indices io, jo. Let c £ M be an element such that

^(eiojo) — cei0j0 — ei0j0c and A(xo) — cxo — xoC.

Put c — Y^ijeS cijeij £ M and a — i=j oijeij + Y1 i£S oiien, where oiien — ciien for every i £ 2.

Lemma 2. Let n be arbitrary different indices, and let b — ij&s bijej £ M be an element such that

A(e%n) — be%n — e%n b and A(xo) — bxo — xob.

c%% - cnn — b%% - bnn

< We have that there exist a, fl such that e%a ,e"n £ {ea3} ( Or ean ,e%" £ {ea/}, or ea 3 £ {eas}), and there exists a diain of pairs of indices (a, fl) in fi, where fi — {(a, fl) : ea / £ {eas}}, connecting pairs (Z, a), (fl,n)> i- e-

(Z,a), (a, Zi), (Zi,m), ..., (m,fl), (fl,v).

Then

c%% — caa — b%% — baa caa — c%l %l — l^a — b%l%l

c%i%i — cnini — b%i%i — bmni cn2n2 — cSS — n2 — bSS cSS — cnn — i/S — ynn

Hence

c%% — b%% — ¿33 — b"" ¿33 — b"" — c%i%i — b%i%i

c%i%i — b%i%i — cnini — bnini, . cn2n2 — bn2n2 — css — bS/3, css — bss — cnn — bnn.

and c%% — b%% — cnn — bnn, c%% — cnn — b%% — bnn. Therefore c%% — cnn — b%% — bnn. >

Lemma 3. Let x be an element of the algebra C(Q, Mn(C)). Then

A(x) — ax — xo,

ao ctS above.

< Let d(ij) £ M be an element such that

A(eij) — d(ij)ej,j — ej d(ij) and A(x) — d(ij)x — xd(ij)

and i = j. Then

A(eij) = d(ij)eij - ejd(ij) = eiid(ij)eij - eijd(ij)ejj + (1 - eii)d(ij)ej - ejd(ij)(1 - ejj) — a(ij )ii eij eij a(ij )jj + ^ ^ a?n e^n eij j ^ a?n e^n

ij Since

eii

d(ij) eij - eij d(ij) ejj = a(ij )ii eij eij a(ij )jj

we have

(1 - eii)d(ij)eii = ^ afne^eii, ejjd(ij)(1 - ejj) = ejj ^ afne^ ij

Let b = ^ijeS bijeij G M be an element such that

A(eij) = beij - eijb and A(xo) = bxo - xob. Then bii - bjj = cii - cjj by Lemma 2. We have bii - bjj = d(ij)ii - d(ij)jj since

beij - eijb = d(ij)eij - eijd(ij).

Hence

cii - cjj = d(ij)ii - d(ij)jj, cjj - cii = d(ij)jj - d(ij)ii. Therefore we have

ejjA(x)eii = ejj(d(ij)x - xd(ij))eii — ejj d(ij)(1 ejj )xeii + ejj d(ij )ejj xeii ejj x(1 eii)d(ij)eii ejj xeiid(ij)eii — ejj ^ ^ ^e^nxeii ejjx ^ ^ a^ne^neii + ejjd(ij)ejjxeii ejjxeiid(ij)eii

— ejj ^ ^ ^e^nxeii ejjx ^ ^ a^ne^neii + ^jjejjxeii ejjxeiic'eii

— ejj ^ ^ ^e^nxeii ejjX ^ ^ a^ne^neii + ejj | ^ ^ j xeii ejjx | ^

— ejj ^ a n e^nxeii ejj x ^^ ^ a ne^neii — ejj (ax xa) eii • ?,nes i,nes

Let d(ii), v, w G M be elements such that

A(eii) = d(ii)eii - eiid(ii) and A(x) = d(ii)x - xd(ii),

A(eii) = veii - eiiv, A(eij) = vej - eijv, A(eii) = weii - eiiw, A(eji) = weji - ejiw.

Then we deduce

(1 - eii)a(ij)eii = (1 - eii)veii = (1 - eii)d(ii)eii, eiia(ji)(1 - eii) = eiiw(1 - eii) = eiid(ii)(1 - eii).

By Lemma 1

A(eij) = a(ij)eij - eja(ij) = E aine(vev - ej E aine(v + a(ij)iiev - eva(ij)33

and

(1 - eu)a(ij)eu = E aineiveu.

Similarly

eu a(ji)(1 - eu) = e^ E ain e(v.

Taking all this into account, we derive the following chain of equalities:

enA(x)en = en(d(ii)x - xd(ii))eu — eiid(ii')(1 eii)xeii + eii d(ii')eii xeii eiix(1 eii ')d(ii')eii eii xeiid(ii')eii — eiia(ji')(1 eii)xeii + eii d(ii')eii xeii eiix(1 eii ')a(ij')eii eii xeiid(ii')eii — eii ^ ^ a^ e£v xeii eiix ^ ^ eii + eiid(ii^eii xeii eiixeiid(ii)eii

= en / a einxea eax / a ein ea

+ cu

eiixeii eiixc eii

— eii^ ^ eiv xeii eiix^~^ a^ ein eii + a^ eii ) xeii eiix | ^ ^ eii

i=n i=n ^ i ' ^ i

— aeinxeii eiix aeineii — eii (ax xa)eii.

It follows that

A(x) = ax - xa

for all x £ C(Q,Mn(C)). >

Proof of Theorem 1. By Lemma 3 A(eii) = aeii - eiia £ M. Hence

y] aiieii - E aiieii £ M. ii

Then

and

aiieii - aiieii = aiieii - aiieii £ M i i i

aiieii - aiieii eii = aiieii - aiieii £ M. i i i

Therefore, ^ a^e^, Y^i aiieii £ M, i. e. aeii,eiia £ M. Hence en ax,xaen £ Mi Let

V = E ^eij : {Aij j C C(X) .

Then ax,xa G V for each element x = {xijeij} G C(Q,Mn(C)), i. e.

x6eij, a6eij G C(Q)ej

£ £

for all i, j. Thus, for all x, y G C(Q, Mn(C)) we have that the elements ax, xa, ay ya, a(x + y), (x + y)a belong to V. Hence

A(x + y) = A(x) + A(y)

by Lemma 3.

Similarly for all x, y G C(Q,Mn(C)) we have

(ax + xa)y = axy - xay G M, axy = a(xy) G V.

Then xay = axy - (ax - xa)y and xay G V. Therefore

a(xy) - (xy)a = axy - xay + xay - xya = (ax - xa)y + x(ay - ya).

Now it can be easily seen that

A(xy) = A(x)y + xA(y)

by Lemma 3. By Section 1 A is homogeneous. Hence, A is a linear operator and a derivation. The proof is complete. >

If we take the *-algebra C(Q,Mn(F)), F = R or H, then we can similarly prove the following theorem.

Theorem 2. Let A he a 2-local derivation on C(Q,Mn(F)). Then A is a derivation. To prove Theorem 2, we need to repeat the proof of Theorem 1 with very minor modification.

Let ^Oj Fej be the following set

Aijeij : (V i, j : Aij G F) (V e > 0) (3 n0 G

(V n ^ m ^ nO)

n

E

i=m

E

Lfc=i,...,i—i

(Akiefci + A e^) + A*e

< e >.

where || ■ || is a norm of a matrix. Then Feij C Mn(F).

Theorem 3. Feij is a C*-algebra with respect to the algebraic operations and the norm in Mn(F) (see [3]).

< We have Fej is a normed subspace of the algebra Mn (F).

Let (an) be a sequence of elements in Feij such th at (an) norm converges to some element a G Mn(F). We have eiianejj ^ eiiaejj at n ^ to for all i and j. Hence eiiaejj G eiiMn(F)ejj for a 11 i, j. Let

bn = E

E(ei-i)i-iaefefe + efcfc aei—M—i) + e^ae*

fe=i

and

£

^ ^(ei— i,i— iamefcfc + efcfcamei—i,i—i) + eiiamei

k=i

Cn =

m

for any ^^en cm ^ 6™ as m ^ to. It should be proven that (6n) is a fundamental sequence. Let e e R+ and fix n. Then there exist m0 such that for all m > mo

yj y^rn

<

Hence for every nO > n and m > mO

E efcfc (b

n — cn )

^mj

. fc=rao

£

- M bn° _ cn° I

b cm

<

3'

At the same time, since am G Yj Fe^, there exists ni > no such that for all l > p > ni we have

Ic1 _ cp I

mm

<

3

Therefore for all l > p > n1 the following relations hold:

lb1 _ &P|| - Mb1 _ ci

cl _ cP I cP _ bP I

mmm

< Mb1 _ ci

+

+llrfA _ ^¡1 < I+1+1=

Since the

Since e is arbitrarily chosen, (6n) is fundamental. Therefore a e Oj eiiMn(F)ejj. sequence (an) is arbitrarily chosen, Fej is a Banach space.

Let ^ jes Si jes 6j be arbitrary elements of the Banach space SOj am = Sw=i aku 6m = Sm=i 6ki natural numbers m. We have the sequence (a m)

converges to je

Let

ajj and the sequence (bm) converges to ^ -eS b

in Fej. Also for

all n and m ambn G SFejj. Then for any n the sequence (ambn) converges to ^ -eS a jj bn

as m ^ to. Hence ^ jeS ajjbn G SOj Fe^. Note that SOj Fej Ç Mn(F). Therefore, for any e G R+ there exi sts nO such that

i,jeE

i , j€E

<

£

j ,j€E

bn+1 _ bn M ^ e

ajjYlj jes bjj as n ^ to.

for any n > nO. Hence the sequence ({aij }6n) converges to Si je Since SOj eiiMn(F)ejj is a Banach space, Si jes aij Si jes 6ij e SOj Feij. Now, the relation SOj Feij C Mn(F), implies that SOj Feij is a C*-algebra. >

Since Feii is a simpie C*-algebra for all i, the proof of Theorem 8 in implies that [3] the C*-algebra SOj Feij is simple.

Let N„(F) = SOj Feij Then C(Q, N^(F)) is a real or complex C*-algebra, where (F = C, R or H) aild C(Q, N„(F)) C M. Hence similar to Theorems 1, 2 we can prove the following theorem.

Theorem 4. Let A be a 2-local derivation on C(Q, N^(F)). Then A is a derivation.

It is known that the set Msa of all self-adjoint elements (i. e. a* = a) of M forms a Jordan algebra with respect to the multiplication a ■ b = ab + ha). The following problem can be similarly solved.

Problem 1: Develop a Jordan analog of the method applied in the proof of Theorem 1 and prove that every 2-local derivation A on the Jordan algebra Msa or C(Q,Mn(F)sa) or C(Q, N„(F)sa) is a derivation.

It is known that the set M& = {a e M : a* = -a} forms a Lie algebra with respect to the multiplication [a, 6] = a6 — 6a. So it is natural to consider the following problem.

e

3

e

e

p

c — c^

mm

a

jj

Problem 2: Develop a Lie analog of the method applied in the proof of Theorem 1 and prove that every 2-local derivation A on the Lie algebra Mk or C(Q,Mn(F)k) or C(Q, Nn(F)k) is a derivation.

The authors thank K. K. Kudaybergenov for many stimulating conversations on the subject.

References

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2. Arzikulov F. N. Infinite order and norm decompositions of C *-algebras I I Int. J. Math. Anal.—2008.— Vol. 2, № 5.-P. 255-262.

3. Arzikulov F. N. Infinite norm decompositions of C*-algebras // Oper. Theory Adv. Appl.—Basel: Springer AG, 2012.—Vol. 220.-P. 11-21. DOI: 10.1186/s40064-016-3468-7.

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Received February 6, 2017 Ayupov Shavkat Abdullayevich

Institute of Mathematics Uzbekistan Academy of Sciences, Director of the V. I. Romanovskiy Institute of Mathematics Do'rmon yo'li Street, Tashkent, 1000125, Uzbekistan E-mail: [email protected]

Arzikulov Farhodjon Nematjonovich Andizhan State University Docent of the Department of Mathematics University Street, Andizhan, 710020, Uzbekistan E-mail: [email protected]

2-ЛОКАЛЬНЫЕ ДИФФЕРЕНЦИРОВАНИЯ НА АЛГЕБРАХ МАТРИЧНО-ЗНАЧНЫХ ФУНКЦИЙ НА КОМПАКТЕ

Аюпов Ш. А., Арзикулов Ф. Н.

В 1997 г. Р. 8етг1 ввел понятие 2-локального дифференцирования и описал 2-локальные дифференцирования на алгебре В(Н) всех ограниченных линейных операторов в бесконечномерном сепара-бельном гильбертовом пространстве Н. После этого, ряд работ был посвящен 2-локальным дифференцированиям на разных типах колец, алгебр, банаховых алгебр и банаховых пространств. Аналогичное описание для конечномерного случая появилось позднее в работе С. О. Кима и Дж. С. Кима. И. Лин и Т. Вонг описали 2-локальные дифференцирования на матричных алгебрах над конечномерным делимым кольцом. Ш. А. Аюпов и К. К. Кудайбергенов предложили новую технику и обобщили упомянутые выше результаты для произвольных гильбертовых пространств. А именно, они рассмотрели 2-локальные дифференцирования на алгебре В(Н) всех линейных ограниченных операторов

в произвольном гильбертовом пространстве Н В(Н)

2-локальным дифференцированиям на ассоциативных алгебрах.

В настоящей работе описаны 2-локальные дифференцирования на различных алгебрах бесконечномерных матрично-значных функций на компакте. Мы развиваем алгебраический подход к исследованию дифференцирований и 2-локальных дифференцирований на алгебрах бесконечномерных матрично-значных функций на компакте и доказываем, что каждое такое 2-локальное дифференцирование является дифференцированием. В качестве основного результата работы установлено, что каждое 2-локальное дифференцирование на *-алгебре С((^, Мп(Е)) или С((^, (Е)), где ^ — компакт, МП(Е) — ^-алгебра бесконечномерных матриц над комплексными числами (вещественными числами или кватернионами), ЛП(Е) — ^-подалгебра в Мп(Е) является дифференцированием. Также поясняется, что разработанный в данной работе метод может быть применен к йордановым и лиевым алгебрам бесконечномерных матрично-значных функций на компакте.

Ключевые слова: дифференцирование, 2-локальное дифференцирование, ассоциативная алгебра, С *-алгебра, алгебра фон Неймана.