ON PRIMITIVE JORDAN BANACH ALGEBRAS MIGUEL CABRERA ...

ON PRIMITIVE JORDAN BANACH ALGEBRAS MIGUEL CABRERA GARCIA, ANTONIO MORENO GALINDO and ANGEL RODRIGUEZ PALACIOS Departamento de An tili3i3 Mat emtitico, Fa cu ltad de Cie n cia3 Univer3idad de Granada, 18071-Granada, Spain

Abstract. We give a description of primitive J ordan Banach algebras J for whi ch there exists an associative primitive algebra A such that J is a Jordan subalgebra of the two-sided Martindale ring of fractions Q s(A) of A containing A as an ideal. Precisely, we prove that there exists a Banach space X and a one-to-one homomorphism ¢ from Q s(A) into the Banach algebra BL(X) of all bounded linear operator on X such that ¢( A) acts irreducibly on X and the restriction of ¢ to J is continuous.

1. Introduction

Since the publication of Zel'manov prime theorem for Jordan algebras [17), several "normed" versions of it have appeared in the literature. Zel'manovian methods have been applied to normed simple algebras with a unit [3], prime J B- and J B* -algebras [6] and non degenerate ultraprime Jordan Banach algebras [4] . Following this line of work, we begin in this note with the consideration of th e Zel'manovian treatment of primitive Jordan Banach algebras in th e way suggested in [13]. The starting point in this direction would be the classification theorem of primitive Jordan algebras , provided independently by A. Anquela, F . Montaner and T. Corts [1] and V. G . Skosyrsky [15]. Acording to this theorem the primitive Jordan algebras over a field K are the following: 1. The simple exceptional 27-dimensional Jordan algebras over a field extension r of K. 2. The Jordan algebras of a nondegenerate symmetric bilinear form on a vector space X over a field extension r of K with dimr(X) 2: 2. 3. Jordan subalgebras of Q~ (A) contai ning A as an ideal , wher e A is a primitive associative algebra over K , 4. Jordan subalgebras of Q.(A) contained in H(Q.(A) , *) and containing H(A , *) as an ideal , where A is a primitive associative algebra over K with a linear algebra involution *. Since the algebras in the case 1 and 2 in the statement have a unit , the field r which arises there can be imbedded into the centre of the algebra. Therefore, as a consequence of the Gelfand-Mazur theorem , all th e complex primitive Jordan Banach algebras in these two first cases are central simple algebras (over C) , and so these algebras are the following: 54 S. Gonzalez (ed.), Non-Associative Algebra and Its Applications, 54-59. © 1994 Kluwer Academic Publishers .

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ON PRIMITIVE JORDAN BANACH ALGEBRAS

1. The simple exceptional 27-dimensional complex Jordan algebra Mf( C) of all hermitian 3x3 matrices over the complex octonions. 2. The Jordan Banach algebras of a continuous nondegenerate symmetric bilinear form on a complex Banach space with dim(X) 2:: 2. Our main result, Theorem 1, gives a precise description of complex primitive Jordan Banach algebras J that are in case 3 in the above theorem, asserting that in such a case J can be seen "well" imbedded in the Banach algebra BL(X) of all bounded linear operators on a suitable Banach space X . A partial result concerning the "purely" hermitian case is also included .

2. The main result We will deal with (linear) Jordan algebras, i. e., algebras over a field of characteristic not two satisfying a.b b.a and the Jordan intentity (a 2 .b).a a 2 .(b.a). We recall that every associative algebra A (whit product denoted by yuxtaposition) gives rise to a Jordan algebra A + under the new product defined by a.b = ~(ab + ba). Subalgebras of A+ are called Jordan subalgebras of A . Also we recall that if A is a prime associative algebra then the Martindale algebra of symmetric quotients of A, Qs(A) , is the maximal algebra extension Q of A satisfying the following conditions: 1. for each q in Q there is a nonzero ideal I of A such that qI and I q are contained in A, and 2. if q is in Q and I is a nonzero ideal of A satisfying ql 0, then q O.

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Theorem 1. Let J be a complex Jordan Banach algebra and assume that there exists a primitive associative algebra A such that J is a Jordan subalgebra of Qs(A) containing A as an ideal. Then there exists a Banach space X and a one-to-one homomorphism ¢ from Qs(A) into the Banach algebra BL(X) of all bounded linear operator on X such that ¢(A) acts irreducibly on X and the restriction of ¢ to J is continuous . A first step for the proof of the theorem is a pur ely algebraic result from which it follows in particular the existence, for each prime associative algebra A , of a nonzero ideal I of A such that z l and I x are contained in A whenever z lies in some Jordan subalgebra of Qs(A) containing A as an ideal. Proposition 1. Let B be an associative algebra and let A be a subalgebra of B. If J is a Jordan subalgebra of B containing A as an ideal, then A 2J ~ A and JA 2 ~ A. Proof. We begin by observing that a 2 x lies in A whenever a is in A and z is in J . This is a consequence of the fact that A is an ideal of J and the equalities a 2x a2.x + Ma2 , x] a 2 .x + [a ,a .x]. By simple linearization, also (a .b)x lies in A whenever a and b are in A and x is in J . Now, to prove A 2 J ~ A , it is enouhg to see that [a, b]x also lies in A . This follows from [a , b]x = [a , b] .x + ~[[a, b], x] = [a, bJ.x + 2(b, x, a)+, where (a, b, c)+ := (a.b).c - a.(b.c). Analogously J A 2 ~ A .

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M. CABRERA, A. MORENO AND A. RODRIGUEZ

Under the assumptions of the above proposition one can consider for each b in A2 the mappings Ab : x -+ bx and Pb : x -+ xb from J into A . Our next goal will be to show that, if J is semiprime and complete normed, then these mappings are continuous. Proposition 2. Let J be a semiprime Jordan Banach algebra, B an associative algebra, A be a subalgebra of B, and assume J can be seen as a Jordan subalgebra of B containing A as an ideal. Then for each b in A 2 the mappings Ab and Pb from J into A defined by Ab(X) := bx and Pb(X) := xb are continuous. Proof. In view of Proposition 1, for b in A 2 , the mapping Db : J -+ B defined by Db ( x) := [b, x] actually is A-valued , hence it is a derivation of the Jordan algebra J. Since D~(x) [b, [b, x]] -4(x , b, b)+ for all x in J, Db is a derivation of the semiprime complete normed algebra J such that Dl is continuous. Since in a semiprime complete normed algebra, a derivation is continuous if its square is continuous [12; Corolario 11.5] we have that Db is continuous. Then it is enough to notice that for all x in J Ab(X) = b.x + ~Db(X) and Pb(X) = b.x - ~Db(X) ,

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Another tool necessary for the proof of Theorem 1 is the following proposition in which we collect in a Jordan context a result whose associative forerunnner is well-know [11; Theorem 2.2.6] . As usual, for a,b,x in a Jordan algebra J we write Ua ,b(X) := a.(x .b) + b.(x.a) - (a.b).x and Ua := Ua ,a . If J is a Jordan sub algebra of an associative algebra, then Ua ,b(X) = ~(axb+bxa) and Ua(x) = axa . Also we recall that an inner ideal of a Jordan algebra J is a subspace I such that UI(J') ~ I where J' is the unital hull of J. An inner ideal I of J is said to be e-modular for e in J if U1_e(J) ~ I, U1-e,I(J') ~ I and e - e2 E I . In such a case the element e is called a modulus for I . We will sayan inner ideal I is e-maximal if it is maximal among all proper e-modular inner ideals. We say I is max imal modular if it is e-rnaxima l for some e. Proposition 3. Let X be a vector space , let J be a Jordan alg ebra of linear operators on X, and assume that J contains as an ideal some (associative) algebra A of linear operators on X acting irredu cibly on X . Then for every nonzero element U in X the linear mapping E u : F -+ F(u) from J into X is onto, and its kernel kerJ(u) := {F E J : F(u) = O} is a maximal modular inner ideal of J with set of modulus id J (11) := {F E J : F( 11) = u} . If moreover we assume that J is a Jordan Banach algebra, then ker J (u) is a closed subspace of J and therefore X becomes in a natural way a Banach space under the norm Ixl := Inf{llFlI : FE J,F(u) = x} . Proof. Fix a nonzero element 11 in X . Since A ~ J and A acts irreducibly on -+ X is onto and idJ(u) is nomempty. It is routine to prove that ker J ( u) is a modular inner ideal of J with set of modulus id J (u). To prove that ker J (11) is a maximal modular inner ideal of J , we fix H in A n id J (u) , we consider an H -rnodular inner ideal P of J containing ker J( u) and we will prove that either P J or P ker J (u). Since A is an ideal of J containing H , P n A is a H-modular inner ideal of A+ containing kerA(u) := {F E A : F(u) = O} . But kerA(u) is a maximal H-modular left ideal of A, hence a maximal H-modular inner

X the linear mapping E; : J

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ideal of A+ ([8; Example 3.3]). It. follows either PnA A or PnA kerA(ll). If PnA = A, then P contains A and hence it also contains H, and so P = J because it does not exclude some of its modula [8; Proposition 3.1]. If PnA = kerA(ll), and if we assume kerJ(u) =f:. P, then we can fix F in P\kerJ(u) and consider G in A such that GF(u) = u. Then FGF(u) = F(GF(u)) = F(u) =f:. 0 and FGF(u) = 0 because Up(A) ~ P n A = kerA(ll), a contradiction . Finally, since maximal modular inner ideals of a Jordan Banach algebra are closed (see [5; Lemma 6.5] or [10; Proposition 6]), if J is a Jordan Banach algebra, then kel'J(ll) is a closed subspace of J, and so , via the canonical linear bijection from J/ kel'J (1l) onto X induced by Eu , X becomes a Banach space under the norm given in the statement . Now we are ready to prove our main result .

Proof of the theorem 1. Since A is a primitive algebra there exists a complex vector space X such that A can be seen as an algebra of linear op erators on X acting irreducibly on X . By [16; Theorem 3.1] also Q$(A) can be seen as an algebra of linear operators on X including A , and so acting irreducibly on X . If we fix a nonzero element u in X, then by Proposition 3 the vector space X can and will be seen as a Banach space for the norm Ixl := Inf{11F11 : F E J , F(ll) = x} . Given G in A 2 and F in J, by Propositions 1 and 2, we have that GF lies in A and IIGFII::; II..\GIIIIFII · Hence , for G in A 2 , x in X , and F in J with F(ll) = z , we have IG(x)1 = IGF(u)1 ::; IIGFII ::; IIAclillFlI , and therefore IG(x)1 ::; IIAclllxl . So every element Gin A 2 is a continuous linear operator on X. Now, given G in Qs(A) we will prove the continuity of G by showing that G has closed graph . By definition of Q$(A), we can choose a nonzero ideal P of A such that PG ~ A, and note that from the irreducible actuation of A on X it follows that p 2 is a nonzero ideal of A acting irreducibly on X . Since p 2 and p 2G are contained in A2 and therefore in BL(X) , given a null sequence {x n } of X such that {G(x n )} --+ y , it follows that for all Fin p2 the sequence {(FG)(x n )} {F(G(x n »)} converges to 0 and to F(y) . Therefore F(y) = 0, and so y O. Now only remains to prove that the embeding (J , II .ID '---> (BL(X) , 1.1) is continx , we have uous . Given Fin kel'](ll), x in X, and G in J with G(u)

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IF(x)1 = I (FG + CF)(u) I :S

IIFC + CFII

= 11 2 F.CII :S 211F11ilCII,

so IF(x)1 :S 211F1l1xi, and so IFI :S 211F11 · Finally we will use this last inequality together with the closed graph theorem to prove the continuity of the embedding (J, 11 .11) '---> (BL(X), 1.1) . Let {Tn} be a null sequence in (J , 11 .11) such that {Tn} 1·1converges to some T in BL(X) . Then , for each G in A 2 such that G(u) = u, the sequence {Tn -TnG} II .II-converges to 0 (Proposition 2) and I.I-converges to T-TG. But, since Tn - TnG lies in ker J (11), we have ITn -1~ GI :S 211Tn - TnGil, and therefore T - TG = 0 for all G in A 2 with G(u) u. Since A 2 acts irreducibly on X, the algebra V of all (possibly discontinuous) linear operators on X which commute with every element in A 2 is a division algebra [2; Proposition 24.6] and a normed algebra for a suitable norm [13; Lemma B.13], hence V ~ C by Gelfand-Mazur 's theorem. Now, for each x in X C-independent with ti, the Jacobson density theorem assures the existence of elements Gin A 2 such that G(u) = u and G(x) = 0, and for such

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M. CABRERA , A. MORENO AND A. RODRIGUEZ

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a G we have T(x) TG(x) T(O) O. It follows that T is zero on X \ Cu. Since obviously we may assume dim(X) ~ 2, we can choose an element x in X C-independent with u. Then , writing u = (u + x) - x , and taking into account that u+x and x both are C-independent with u, we have also T( u) = T( u + x) - T( x) = O. Therefore T = O.

3. A partial result concerning the " p u r ely " hermitian case Recall that an associative algebra with involution (A, *) is said to be a *-tight envelope of a Jordan algebra J includ ed in H(A , *) if A is generated by J and every nonzero *-ideal of A meets J . Theorem 2. Let J be a complex Jordan Banach algebra, and assume that there exists a complex primitive associative algebra A with an algebra involution * such that J = H(A,*) and(A ,*) is a *-tight envelope ofJ . Then there exists a Banach space X such that A can be seen as an algebra of bounded linear operators acting irreducibly on X and the embedding J '---+ BL(X) is continuous. Proof. Since J is a semiprime Jor dan Banach algebra and (A , *) is a *-tight envelope of J, by [14; Theorem 2] (up to equivalent renorming of J ifnecesary) there exists an algebra norm 11.11 on A which extends th e norm of J , makes * isometric , and, if (A,*) denotes the II .II-completion of (A ,*), t hen J = H( A ,*) and every nonzero *-ideal of A meets J. First we note that , if A is commutative, th en A is a division algebra, so A equals C by th e Gelfand-Mazur theorem , and so t he th eorem is true in this case. Assume that A is not commutative. Since A is generated by J and J = H(A, *), A is a Lie ideal of A [7; Example 2 in page 59]. On th e other hand the product of two arbitrary nonzero *-ideals of A is nonzero (hence A is semiprime) because every nonzero *-ideal of A meets H(A,*) and H(A ,*) is a prime Jordan algebra. It follows from [7; Th eorem 2.1.2] that A contains a nonzero ideal P of A. Replacing P by P + P* if necessary we can assume that P is a nonzero *-(hence essential) ideal of A. Then we can identify the Martindale algebra of symmetric quotients of P, with that of A and A [9; Proposition 3.1]. Now, if we see A as an algebra of operators acting irredu cibly on a suitable complex vector space X , th en by [16] A can also be seen as an algebra of operators acting irredu cibly on X . Since Ais a Banach algebra, by [11; Th eorem 2.2.6] X appea rs as a Banach spac e in such a way that all the elements in A are bounded operators and th e embedding of A into BL(X) is continuous.

Acknowledgements The authors want to tank J . Martin ez for several int eresting suggestions concerni ng the topics in this note .


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