Reversible AJW-algebras Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2016, Том 18, Выпуск 3, С. 15-21

yffK 517.98

REVERSIBLE AJW-ALGEBRAS

Sh. A. Ayupov, F. N. Arzikulov

The main result states that every special AJW-algebra can be decomposed into the direct sum of totally irreversible and reversible subalgebras. In turn, every reversible special AJW-algebra decomposes into a direct sum of two subalgebras, one of which has purely real enveloping real von Neumann algebra, and the second one contains an ideal, whose complexification is a C*-algebra and the annihilator of this complexification in the enveloping C*-algebra of this subalgebra is equal to zero.

Mathematics Subject Classification (2000): 17C65, 46L57. AJW-algebra, reversible AJW-algebra, AW*-algebra, Enveloping C*-algebra.

1. Introduction

This article is devoted to abstract Jordan operator algebras, which are analogues of abstract W*-algebras (AW*-algebras) of Kaplansky. These Jordan operator algebras can be characterized as a JB-algebra satisfying the following conditions

(1) in the partially ordered set of all projections any subset of pairwise orthogonal projections has the least upper bound in this JB-algebra;

(2) every maximal associative subalgebra of this JB-algebra is generated by it's projections (i. e. coincides with the least closed subalgebra containing all projections of the given subalgebra).

In the articles [3, 4] the second author introduced analogues of annihilators for Jordan algebras and gave algebraic conditions equivalent to (1) and (2). Currently these JB-algebras are called AJW-algebras or Baer JB-algebras in the literature. Further, in [5] a classification of these algebras has been obtained. It should be noted that many of facts of the theory of JBW-algebras and their proofs hold for AJW-algebras. For example, similar to a JBW-algebra an AJW-algebra is the direct sum of special and purely exceptional Jordan algebras [5].

It is known from the theory of JBW-algebras that every special JBW-algebra can be decomposed into the direct sum of totally irreversible and reversible subalgebras. In turn, every reversible special JBW-algebra decomposes into a direct sum of a subalgebra, which is the hermitian part of a von Neumann algebra and a subalgebra, enveloping real von Neumann algebra of which is purely real [6, 7]. In this paper we prove a similar result for AJW-algebras, the proof of which requires a different approach. Namely, we prove that for every special AJW-algebra A there exist central projections e, /, g e A, e+/+9 = 1 such that (1) eA is reversible and there exists a norm-closed two sided ideal /of C*(eA) such that eA = ±(±(/sa)+)^^2) /A is reversible and R*(/A) n iR* (/A) = {0}; (3) gA is a totally nonreversible AJW-algebra.

© 2016 Ayupov Sh. A., Arzikulov F. N.

(2) for every subset S C A+ there exists a projection e e A such that (S= Ue(A);

(3) for every subset S C A there exists a projection e e A such that ±(S)+ = Ue(A+) [3].

2. Preliminary Notes

We fix the following terminology and notations.

Let A be a real Banach *-algebra. A is called a real C*-algebra, if Ac = A + %A = {a + ib : a,b e A}, can be normed to become a (complex) C*-algebra, and keeps the original

A

Let A be a JB-algebra, P (A) be a set of all project ions of A. Further we will use the following standard notations: {aba} = Uab := 2a(ab) — a2b, {abc} = a(bc) + (ac)b — (ab)c and {aAb} = {{acb} : c e A}, where a,b,c e A. A JB-algebra A is called an AJW-algebra, if the following conditions hold:

P(A) A

(2) every maximal associative subalgebra Ao of the algebra A is generated by it's projections (i.e. coincides with the least closed subalgebra containing Ao n P(A)). Let

(S)x = {a e A : (Vx e S) Uax = 0}, X(S) = {x e A : (Va e S) Uax = 0}, ±(S)+ = ±(S) n A+.

A

A

1+

--------------„ e , „.„u .u±(S} +

Let A be a real or complex *-algebra, and let S be a nonempty subset of A. Then the set R(S) = {x e A : sx = 0 for all s e S} is called the right annihilator of S and the set L(S) = {x e A : xs = 0 for all s e S} is called the left annihilator of S. A ^^^^bra A is called a Baer if ^te right annihilator of any nonempty set S C A is generated

by a projection, i.e. R(S) = gA for some projection g e A (g2 = g = g*). If S = {a} then the projection 1 — g such that R(S) = gA is called the right projection and denoted by r(a). Similarly one can define the left projection I (a). A (real) C*-algebra A, which is a Baer (real) *-algebra, is called an (real) AW*-algebra [1, 8]. Real AW*-algebras were introduced

*A

C^^^^bra M = A + iA is not necessarily a complex AW*-algebra.

Let A ^e an AJW-algebra. By [5, Theorem 2.3] we have the equality A = Aj © Ajj © A///,

where A/ is an AJW-algebra of type I, A// is an AJW-algebra of type II and Aj/j is an AJW-

Aj

form

Aj = A^ © Ai © A2 © ...,

where An for every n is {0} or an AJW-algebra 0f type In A^ is a direct sum of AJW-

algebras of type Ia with a infinite. If A = A1 © A2 © ... then A is called an AJW-algebra of type Ifn and denoted by A/fin and if A = A^ then A is called an AJW-algebra of type

and denoted by A/^. We say that A is properly infinite if A has no nonzero central modular

Ajj

to JBW-algebras [9]. Therefore, it is isomorphic to some AJW-algebra defined in [14] (i.e. to some AJW-algebra of self-adjoint operators), and by virtue of [14] Aj/ = Aj/l © A//tc, where Aj/l is a modular AJW-algebra of type II and A//tc is an AJW-algebra of type II, which is properly infinite. So, we have the decomposition

A = Aj © A/x © A//i © A//x © A///.

It is easy to verify that the part AjJin ® Ajj1 is modular, and Aj^ ® Ajj^ ® Am is properly infinite (i.e. properly nonmodular).

3. Reversibility of AJW-algebras

Let A be a special AJW-algebra on a complex Hilbert space H. By R*(A) we denote the uniformly closed real *-algebra in B (H), generated by A, and by C*(A) the C*-algebra, generated by A. Thus the set of elements of kind

n mi

Y^ n an (an£ A) i=i j=i

is uniformly dense in R*(A). Let iR*(A) be the set of elements of kind ia, a £ R*(A). Then C*(A) = R* (A) + iR* (A) [7, 13].

Lemma 3.1. The set R*(A) n iR* (A) is a uniformly closed two sided ideal in C*(A).

< If a, b £ R*(A) an dc = id £ R*(A)niR*(A), then (a+ib)c = ac+ibid = ac-bd £ R*(A). Similarly (a+ib)c £ iR*(A),i.e. (a+ib)c £ R*(A)niR*(A). Sinee R*(A)niR*(A) is uniformly closed and the set of elements of kind a + ib, a, b £ A is uniformly dense in C*(A), we have R*(A) n iR*(A) is a left ideal in C* (A). By the symmetry R*(A) n iR* (A) is a right ideal. >

Let R be a *-algebra, Rsa be the set of all self-adjoint elements of R, i. e. Rsa = {a £ R : a* = a}.

Definition 3.2. A JC-algebra A is said to be reversible if a1a2 ... an + anan-1 ...a1 £ A for all a1} a2,... ,an £ A.

Similar to JW-algebras we have the following criterion.

Lemma 3.3. An AJW-algebra A is reversible if and only if A = R*(A)sa.

< It is clear that, if A = R*(A)sa, then A is reversible since

n 1 \ * 1 n

ai + ai i ^ \ai +

=1 i=n / i=n i=1

]Jai ^fj aA ^JJ ai ^ü ai c R*(A)sa = A,

for all a1,a2,... ,an £ A. Conversely, let A be a reversible AJW-algebra. The inclusion A c R*(A)sa is evident. If a = En=^m=1 aij £ R*(A)sa, then

! ! n / mi 1 x

a = - {a + a*) = - ^ i J] aij + J] aij J £ A.

i=1 j=1 j=mi

Hence the converse inclusion holds, i.e. R*(A)sa = A >

Lemma 3.4. Let A be an AJW-algebra and let I be a norm-closed ideal of A. Then there exists a central projection g such that ±(±(Isa)+)+ = gA+.

< Since A is an AJW-algebra there exists a projection g in A such that

±(Isa)+ = U(1-g) (A+), ^ (±(Isa) +)+ = Ug (A+),

where ±(S)+ = {x £ A+ : (V a £ S) Uax = 0} for S C A.

Let (u\) ^e an identity of the JB-subalgebra I and a be an arbitrary positive

element in I. Then there exists a maximal associative subalgebra A0 of A containing a. Let

v^ be an approximate identity of Aa. Then (v^) C (uA^d \\av^ — a\\ ^ 0. Let b £ A+ and UvMb = 0 for every y. Then UaUvMb = UaVjlb = 0 mid UcUavMb = 0 where c is an element in A such that b = c2. Hence UcUavMc2 = 0 (Uc(av^))2 = 0 Uc(av^) = 0 and UcUc(av= Ub(av^) = 0 for every y. We have

\\Ub(av^) — Uba\\ = \\Ub(av^) — a)\\ ^ 0

because \\av^ — a\\ ^ 0 and the operator Ub is norm-continuous. Hence Uba = 0. We may assume that a = d2 for some element d £ A. Then

Ud Uba = Ud Ubd2 = (Ud b)2 = 0, Ud b = 0.

Thus UdUdb = Ud2b = Uab = 0. Therefore, if b £x ((uA))+ then b £x (Isa)+. Hence x((ua))+ (Isa)+. ft is clear that x(Isa)+ ((uA))+ and

±(Isa)+ = ((ux))+.

This implies that x((ux))+ = U(ï-g) (A+) and

sup u\ = g. x

Let us prove that Ug(A) is an Weal of A. Indeed, let x be an arbitrary element in A. Then Uxux G Isa, i.e. Uxux G Ug(A). By [9, Proposition 3.3.6] and the proof of [9, Lemma 4.1.5] we have Ux is a normal operator in A. Hence

At the same time

sup Uxux = UxXsup ux) = Uxg. xx

sup Uxux G Ug (A). x

Hence Uxg £ Ug(A). By [9, 2.8.10] we have

4(xg)2 = 2gUx g + Uxg2 + Ug x2 = 2gUxg + Uxg + Ug x2.

Therefore (xg)2 £ Ug(A) and xg £ Ug(A).

Now, let y be an arbitrary element in UgA. Then y = Ugy and

xy = (Ug x + {gx(l — g)} + Ui-g x)Ug y = Ug xUg y + {gx(l — g)} Ug y £ Ug A

since {gx(l — g)} £ UgA. Hence UgA is a norm-closed ideal of A. Therefore {gA(l — g)} = {0} and

A = Ug A © Ui-g A. This implies that g is a central projection in A mid ±(±(Isa)+)+ = gA+. >

Lemma 3.5. Let A be a reversible AJW-algebra on a Hilbert space H. Then there exist two central projections e, f in A and a norm-closed two sided ideal I of C*(A) such that e + f = 1, eA = ±(±(Isa)+)+ and R*(fA) n iR*(fA) = {0}.

< Let I = R*(A) n iR*(A). Sinee A is reversible by Proposition 3.3 we have Isa C A. By [7, 3.1] I is a two sided ideal of C* (A). Hence Isa is an ideal of the AJW-algebra A. By Proposition 3.4 we have there exists a central projection g such that x(x(Isa)+)+ = gA+. It is clear that g is a central projection also in C*(A).

Ig

R*((1 — g)A) n iR*((1 — g)A) = {0}. >

Lemma 3.6. Let A be an AJW-algebra and let J be the set of elements a £ A such that bac + c*ab* £ A for all b,c £ C*(A). Then J is a norm-closed ideal in A. Moreover J is a reversible AJW-algebra.

< Let a, b £ J, s,t £ C*(A). Then

s(a + b)t + t*(a + b)s* = (sat + t*as*) + (sbt + t*bs*) £ A,

i. e .J is a linear subspace of A. Now, if a £ J b £ A, s,t £ C *(A), then

s(ab + ba)t + t*(ab + ba)s* = (sa(bt) + (bt)*as*) + ((sb)at + t*a(sb)*) £ A,

i. e. J is a norm-closed ideal of A.

Let a1 £ J a2,... ,an £ A and a = nn=2 a%. Then a1 a + a*a1 £ A % the definition of J. Let us show that a1a + a*a1 £ J; then, in particular, in the case of a2,..., an £ J this will imply that J is reversible. For all b, c £ C* (A) we have

b(a1 a + a*a1)c + c*(a1 a + a*a1 )b* = (ba1(ac) + (ac)*a1b*) + ((ba*)a1 c + c*a1(ba*)*) £ A,

i. e. a1a + a*a1 £ J. >

Definition 3.7.An AJW-algebra A is said to be totally nonreversible, if the ideal J in {0} J = {0}

Theorem 3.8. Let A be a special AJW-algebra. Then there exist central projections e, f,g £ A e + f + g = 1 suc^ that

(1) J = (e + f )A, J is the ideal from Lemma 3.6;

(2) eA is reversible and there exists a norm-closed two sided ideal I of C*(eA) such that

eA = HHha)+)+;

(3) f A is reversible and R*(fA) n iR* (f A) = {0};

(4) gA is a totally nonreversible AJW-algebra and

gA C(Qu, R ® Hu),

where ft is a set of indices, {Q^ is an appropriate family of extremal compacts and {Huis a family of Hilbert spaces. < We have

A = A1 ® A2 ® ■ ■ ■ ® Ajx ® Ajji ® AJJx ® AJJJ and the subalgebra (without the part A2)

A1 ® A3 ® A4 ® ■ ■ ■ ® AJx ® AJJi ® AJJx ® AJJJ

is reversible. The last statement can be proven similar to [9, Theorem 5.3.10]. By [10] the subalgebra A2 can be represented as follows

A2 = C(Xi, R ® Hi),

where 2 is a set of ind ices, [Xi }ies is a family of extremal comp acts and [Hi }ies is a family of Hilbert spaces. Hence by [9, Theorem 6.2.5] there exist central projections h, g such that A = hA © gA hA is reversible and gA is totally nonreversible. For all a b\,... ,bn,c\,cm in hA we have

bi... b,naci ...cm + cmcm-1 ... ciabnbn-1 ...bi £ hA

since hA is reversible .Similarly for all b,c £ R*(hA)7 a £ hA we have

bac + c*ab* £ hA.

Hence hA = J.

e f hA

sided ideal I of C*(hA) such that e + f = h, eA = x(x(Isa)+)+, f A is a reversible AJW-algebra and R*(fA) n iR*(fA) = {0}. This completes the proof. >

Let A be a special AJW-algebra. Despite the fact that for the real AW*-algebra R*(A) the C*-algebra M = R*(A) + iR*(A) is not necessarily a complex AW*-algebra we consider, that

Conjecture. Under the conditions of Theorem 3.8 the following equality is valid

eA = Isa-

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0

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Received September 24, 2015.

Ayupov Shavkat Abdullayevich

National University of Uzbekistan,

Director of the Institute of Math.

Do'rmon yo'li St., Tashkent, 1000125, UZBEKISTAN

Email: [email protected]

Arzikulov Faehodjon Nematjonovich Andizhan State University, Department of Mathematics docent, University street, Andizhan, 710020, UZBEKISTAN Email: [email protected]

ОБРАТИМЫЕ АШ-АЛГЕБРЫ Аюпов Ш. А., Арзикулов Ф. Н.

Основной результат статьи гласит, что каждая специальная А1"\¥-алгебра раскладывается в прямую сумму тотально необратимой и обратимой подалгебр. В свою очередь, каждая обратимая А1"\¥-алгебра раскладывается в прямую сумму подалгебры, которая содержит идеал такой, что аннулятор комплексификации этого идеала в обертывающей С *-алгебре этой подалгебры равен нулю и подалгебры, обертывающая вещественная алгебра фон Неймана которой является чисто вещественной.

Ключевые слова: АД^^-алгебра, обратимая АД^^-алгебра, ЛШ*-алгебра, обертывающая *-алгебра.