Reversible AJW-algebras

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).

• 2.    Preliminary Notes

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. f. g E A. e +f+ g = 1 such lltat (1) eA is reversible and there exists a norm-closed two sided ideal I of C *(eA) such that eA = ^±(^±(I_sa)+)+; (2) fA is reversible and R*(fA) П iR* (fA) = {0}; (3) gA is a totally nonreversible AJW-algebra.

We fix the following terminology and notations.

Let A be a rea 1 Banach *-algebra. A is called a real C*-algebra. if Ac = A + iA = {a + ib : a, b E A }, can be normed to become a (complex) C*-algebra, and keeps the original norm on A [11].

Let A be a JB-algebra. P (A) be a sei, of all projections of A. Further we will use Ilie 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:

• (1)    in the partially ordered set P (A) of projections any subset of pairwise orthogonal projections has the least upper bound in 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 П P (A)).

Let

(S ^ = {a e A : (Vx E S) Uax = 0},

^±(S) = {x E A : (V a E S) Uax = 0},

±(S)+ = ±(S) П A+.

Then for a JB-algebra A the following conditions are equivalent:

• (1)    A is an AJW-algebra:

• (2)    for every subset S C A+ there exists a projection e E A such that (S)^± = U_e(A):

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

• 3.    Reversibility of AJW-algebras

Let A be a. real c>r complex *-algebra. arid let S Ire a nonempty subset of A. Then the set R(S) = {x E A : sx = 0 for all s E S} is called the riglit annihilator of S arid the set L(S) = {x E A : xs = 0 for all s E S} is called the left annihilator of S. .X *-algebra, A is called a Baer *-algebra. if the right arniiliilai,or 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 I,lie left projection l(a). A (rA the C*-a,lgebra, M = A + iA is riot necessarilу a complex AA’*-algebra.

Let A l_>e an AJA'-algebra. By [o. Theorem 2.3] we have the equality A = Ai ф Aii ф Am. where Ai is an AJW-algebra, of type I. Aii is an AJW-algebra, of type II arid Аш is an AJA'-algebra of type III [5]. By [5, Theorem 3.7] Ai, in its turn, is a direct sum of the following form

Ai = A^ ф Ai ф A2 ф ..., where An for every n eitlier is {0} or an AJW-algebra of type In. A^ is a direct sum of AJA"-algebras of type Ia with a infinite. If A = A1 ф A2 ф ... then A is called an AJW-algebra of type I fin and denoted by Afin and if A = A^ then A is called an AJW-algebra of type 1^ and denoted by Ai^. We saу that A is properly infinite if A has no nonzero central modular projection. The fact that an AJW-algebra Aii of type II is a JC-algebra can be proved similar 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] Aii = An1 ф An^, where Aii1 is a modular AJW-aIgebra of type II and aii^ is an AJW-algebra of type II, which is properly infinite. So, we have the decomposition

A = A i fin ф Ai^ ф Aii1 ф Aii^ ф Лщ .

It is easy to verify that the part Afin ф Aii1 is modular, and Aj^ ф Aii^ Ф Aiii is properly infinite (i. e. properly nonmodular).

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

52 П “j ⁽aij ^{E A)}

i=1 j=1

is uniformly dense in R*(A). Lel, iR*(A) l>e the set of elements of kind ia. a E R*(A). Then C*(A) = R*(A) + iR* (A) [7. 13].

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

C I fa, b E R*(A) aiid c = id E R*(A)CiR*(A). then (a+ib)c = ac+ibid = ac-bd E R*(Ay Similarly (a + ib)c E iR*(A). i. e. (a+ib)c E R*(A)ПiR*(A). Siiice R*(A)ПiR*(A) is uniformly closed and the set of elements of kind a + ib, a,b E A is uniformly dense in C * (A), we have R*(A) П iR*(A) is a left ideal in C *(A). By the syinmetry R*(A) П iR* (A) is a riglit ideal. B

Let R be a *mlgebra. R_sa be the set of all self-adjoint elements of R. i.e. R_sa = {a E R : a* = a}.

Definition 3.2. A JC-algebra A is said to be reversible if a₁a₂ .. .an + a_na_n-1... a₁ E A for all ai, a2,..., an E 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.

C II, is clear that, if A = R*(A)sa. then A is reversible since

• ( П ai + П “i] ’ = П ai + П ai E R4A)«, = A, ^x i=1 i=n ' i=n i=1

for all a₁,a₂,... ,a_n E A. Conversely, let A lie a reversilrle AJW-algelira. The inclusion A C R*(A)sa is evid ent. If a = Pn₌₁ Oj=_i “ij E R*(A)sa. then

“ =2(“ + a’) = 2 X ( П “j + П “■/) E A-

• i =1 j =1       j = m i

Hence the converse inclusion holds, i.e. R*(A)_sa = A. B

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 ^(^(I_sa )+)+ = gA+-

C Siiice A is an AJW-algelira there exists a projection g in A such that

±(Isa)+ = U(1-g) (A+))   ±(±(Isa) + )+ = Ug(A+)> where ±(S)+ = {x E A+ : (Va E S) Uax = 0} ft>r S C A.

Let (u\) be an approximate ideiitity of Ilie JB-subalgebra. I arid a l>e an arlaitrary positive element in I. Then there exists a iiiaximal associative sul>algel>ra Ao оf A coiitaiiiiiig a. Let vg be an approxiniate identity of Ao. Tlien (v^) C (ид) arid ||avg — ak ^ 0. Lel, b E A+ and UvMb = 0 for every p. Then UaUv^b = UaVpi b = 0 arid UcUav^b = 0. where c is an element in A such that b = c2. He nee UcUav^ c = 0, (Uc(avg))2 = 0, Uc(av^) = 0 and UcUc(avg) = Ub(avg) = 0 for every p. We have

• kUb (av^) - Ubak = ||Ub(av_g) — a)k ^ 0

because ||av_g — ak ^ 0 and tire operator Ub is norm-continuous. Hence Uba = 0. We may assume that a = d² for some element d E A. Then

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

Thus U_dU_d b = U_d2 b = Uab = 0. There fore, if b E^ ((ид))+ theii b E^ (I_sa )+• Hence ^± (( ua ))+ C^± (Isa )+• It is cl ear that ^±(Isa)₊ C^± ((ид))₊ and

±(Isa)+ =± ((ид)) +.

This implies that ±((ид))+ = U(1-g)(A+) and sup ид = g. λ

Let us prove that Ug(A) is an i(leal of A. Indeed, let x be an arbitrary element in A. Then Uxuд E Isa, i. e. Uxuд E 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 sup Uxuд = Ux(sup ид) = Uxg. λλ

At the same time

SUP UxUД E Ug (A). λ

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

• ^4(xg)2 = ^2gUx^{g + U}x^{g2 + U}gx = ^2gUx^{g + U}x^{g + U}gx •

Therefore (xg)² E Ug (A) ar id xg E Ug (A).

Now. let y lie an arbitrary element in UgA. Then y = Ugy and xy = (Ugx + {gx(1 — g)} + Ui-gx)Ugy = UgxUgy + {gx(1 — g)}Ugy E UgA since {gx(1 — g)} E Ug A. Hence Ug Ais a norm-closed ideal of A. Thei-efore {gA(1 — g)} = {0} and

A = Ug A ф U1-g A.

This implies that g is a central projection in A arid ^±(^±(I_sa)+)+ = gA+. B

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

C Lel, I = R*(A) П iR*(A). Siiice A is rer'ersible l>y Proposition 3.3 we Inwe Isa C A. By [7. 3.1] I is a l,wo sided ideal of C *(A). He nee Isa is an ideal of ll ie AJW-algebra A. By Proposition 3.4 we have there exists a central projection g such that ^±(^±(I_sa)+)+ = gA+. Il, is clear that g is a central projection also in C *(A).

By the dofiiiil ions of I and g wo Inrco

R*((1 — g)A) П iR*((1 — g)A) = {0}. ▻

Lemma 3.6. Let A be an AJW-algebra and let J be the set of elements a E A such that bac + c*ab* E A for all b,c E C *(A). T1 ten J is a norm-closed ideal in A. Moreover J is a reversible AJW-algebra.

C Lel, a, b E J. s,t E C *(A). Then

s(a + b)t + t*(a + b)s* = (sat + t*as*) + (sbt + t*bs*) E A, he. J is a linear subspace of A. Now. if a E J. b E A. s,t E C*(A). then

s(ab + ba)t + t*(ab + ba)s* = (sa(bt) + (bt)*as*) + ((sb)at +t* a(sb)*) E A, i.e. J is a norm-closed ideal of A.

Let ai E J, a2,..., an E A and a = ЦП=2 ai. Tlieii aia + a*ai E A by I,he definition of J.

Lei us show that aia + a*ai E J. then, in particular, in I,he ease of a2,...,a_n E J this will imply I,ha I, J is reversible. For all b. c E C *(A) we have

b(a1 a + a*a_i)c + c*(a_i a + a*a_i )b* = (ba_i(ac) + (ac)*a_ib*) + ((ba*)a_i c + c* a_i(ba*)*) E A,

• i.    e. aia + a*ai E J. B

Theorem 3.8. Let A be a special AJW-algebra. Then there exist central projections e, f,g E A. e + f + g = 1 such 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 = Wsn)+)+;

• (3)    fA is reversible and R*(fA) П iR* (fA) = {0};

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

gA = X C (Q_Ш, R ф Нш ),

ω∈Ω where D is a set оf indices, {QШ }wGq is an appropriate family of extremal compacts and {Нш}шей is a family of Hilbert spaces.

C We haa’c

A = Ai ф A2 ф ••• ф Ai^ ф Ani ф An^ ф Am and the subalgebra (without the part A2)

A1 ⊕ A3 ⊕ A4 ⊕ · · · ⊕ AI∞ ⊕ AII1 ⊕ AII∞ ⊕ AIII 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 = X C(Xi, R ф Hi), i∈Ξ where S is a set оf indices. {Xi}ie= is a family of extrernal compacts and {Hi}ie= is a family of Hilbert spaces. Hence by [9, Theorem 6.2.5] there exist central projections h, g such that A = hA ф gA. hA is rer'ersible and gA is totally iionrer'ersible. For all a. bi,..., bn, ci,... ,cm in hA we have bi . . . bnaci . . . cm + cmcm—i . . . ciabnbn—i . . . bi E hA since hA is reversible. Similarly for all b,c E R*(hA). a E hA wo have bac + c*ab* E hA.

Hence hA = J.

By Proposition 3.5 there exist two central projections e, f in hA and a norm-closed two sided ideal I of C*(hA) such that e + f = h. eA = ^±(^±(I_sa)+)+• fA is a reversible AJW-algebra and R*(fA) П iR*(fA) = {0}. This complei,es the proof. B

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

CONJECTURE. Under the conditions of Theorem 3.8 the following equality is valid eA = Isa.