Jordan Ideals Revisited

Monatsh. Math. 145, 1–10 (2005)

DOI 10.1007/s00605-004-0279-7

Jordan Ideals Revisited

By

Matej Bresar1, Ajda Fosner1, and Maja Fosner2

1 University of Maribor, Slovenia2 Institute of Mathematics, Ljubljana, Slovenia

Received October 15, 2003; accepted in revised form January 14, 2004Published online November 4, 2004 # Springer-Verlag 2004

Abstract. An alternative approach to the study of Jordan ideals in associative algebras is considered.The same method can be used when analogous questions in graded algebras are treated.

2000 Mathematics Subject Classification: 16W10, 16W50, 16W55, 17C50, 46H10Key words: Jordan ideal, associative algebra, "-Jordan ideal, graded algebra, involution

1. Introduction

Let A be an associative algebra over a commutative ring �. It is well-knownthat A becomes a Jordan algebra (resp. Lie algebra) if we replace the originalproduct by the Jordan product a � b ¼ abþ ba (resp. Lie product ½a; b� ¼ab� ba). Recall that a �-submodule J of A is called a Jordan ideal of A ifa � x2J for all a2A, x2J. The standard problem, which has been studied formore than 50 years, is to describe Jordan ideals in terms of (ordinary) ideals, that isto find conditions on A under which every nonzero Jordan ideal of A is in fact anideal, or at least it contains a nonzero ideal. Fundamental results in this area aredue to Herstein (see e.g. [12]). Our aim is to point out some entirely elementaryobservations, which nevertheless yield some new information on Jordan ideals.Whereas, generally speaking, this paper is written in the spirit of Herstein’s clas-sical works, the main idea of our approach is different. It is based on the followingwell-known identity connecting the Jordan and the Lie product:

½½a; b�; c� ¼ a � ðb � cÞ � b � ða � cÞ: ð1ÞThe core of the paper is Section 2 where Jordan ideals of algebras are treated.

From the basic result, Theorem 2.1, we shall, on the one hand, derive short proofs ofseveral known theorems, and on the other hand we shall obtain some new results;among them we mention the one stating that ideals are the only Jordan ideals of vonNeumann algebras. Our approach is also applicable to the study of Jordan ideals ofsymmetric elements in algebras with involution, though it yields less ultimateresults (Section 3). Moreover, modifications of our methods work in the context

Partially supported by a grant from M�SSZ�SS.

of graded algebras. In Section 4 we shall consider "-Jordan ideals of G-gradedalgebras; here, G is an abelian group and " is a bicharacter for G. In particular,we shall extend some results from the recent papers by Bergen and Grzeszczuk [1]and G�oomez-Ambrosi and Montaner [9]. Finally, Section 5 gives a result on "-Jordanideals of symmetric elements in graded algebras with color involution.

Let us fix our notation. Throughout, � will be a commutative unital ringcontaining the element 1

2(i.e., 1 þ 1 is invertible in �). For the sake of simplicity

of the exposition we have decided to work in the context of �-algebras instead ofthat of 2-torsionfree rings. Though our arguments could be adapted to that setting,some of the statements would then become somewhat more complicated. Thus, byan algebra we shall always mean an associative algebra over �. If S;T aresubsets of an algebra A, then by ½S;T� we denote the �-submodule generatedby all ½s; t�, s2S, t2T. Analogously we introduce ST, S �T etc. By AlgðSÞ(resp. IdðSÞ) we denote the subalgebra (resp. ideal) of A generated by S. Thecenter of A will be denoted by Z.

2. Jordan Ideals of Algebras

Theorem 2.1. If J is a Jordan ideal of an algebra A, then J contains the setAlgð½A;A�ÞJ. In particular, J contains the ideal Idð½½A;A�;A�ÞJ.

Proof. Let a; b2A and x2J. Using (1) (with c ¼ x) we see that ½½a; b�; x� 2J.On the other hand, ½a; b� � x2J. From both relations one readily concludes that½a; b�x2J, which in turn yields Algð½A;A�ÞJ � J. The other relationIdð½½A;A�;A�ÞJ � J is a direct consequence since

Idð½½A;A�;A�Þ � Algð½A;A�Þ: ð2ÞTo prove this we first note that

a½½b; c�; d� ¼ ½a½b; c�; d� � ½a; d�½b; c�;which shows that A½½A;A�;A� � Algð½A;A�Þ. Similarly, ½½A;A�;A�A �Algð½A;A�Þ. Finally, if a; b2A and u2 ½½A;A�;A�, then aub ¼ ½au; b� þðbaÞu2Algð½A;A�Þ, proving (2). The set Idð½½A;A�;A�ÞJ is indeed anideal of A. It is trivially a left ideal, and using vxa ¼ vðx � aÞ � ðvaÞx withv2 Idð½½A;A�;A�Þ, x2J, a2A, we see that it is also a right ideal. &

By the very definition of a Jordan ideal we see that the conclusion of Theorem2.1 implies that J also contains JAlgð½A;A�Þ and JIdð½½A;A�;A�Þ; the latterset is of course an ideal. Therefore, the following corollaries are immediate.

Corollary 2.2. LetA be such that Algð½A;A�Þ ¼ A. Then every Jordan idealof A is an ideal.

Corollary 2.3. Let A be such that Idð½½A;A�;A�Þ has either trivial rightannihilator or trivial left annihilator. Then every nonzero Jordan ideal of Acontains a nonzero ideal.

One can compare our conclusions with the one obtained by Herstein [10]. Heproved that a Jordan ideal J of A contains the ideal IdðJ �JÞ (to be precise, this

2 M. Bre�ssar et al.

is not explicitly written in Herstein’s paper, but can be deduced by inspecting hisproof, cf. [16]). Therefore, in order to use Herstein’s construction one is forced toanalyse the set J �J (which can be 0 in the worst case). On the other hand, theabove corollaries immediately reduce the question on the structure of Jordan idealsof A to the questions on some structural properties of A. The first question is:when is Algð½A;A�Þ equal to A? It was Herstein who proved that this is true if Ais a noncommutative simple algebra (see e.g. [12, p. 6]). An alternative proof,which avoids applying the results on Lie ideals, can be easily established: just use(2) and conclude from the next simple lemma that ½½A;A�;A� 6¼ 0 if A is anoncommutative simple algebra.

Lemma 2.4. Let A be such that the center Z of A does not contain nonzeronilpotent elements. If c2A satisfies ½½A; c�;A� ¼ 0, then c2Z.

Proof. Our assumption can be read as ½A; c� � Z. Let a2A. Since ½a; c� 2Zit follows that ½a; c�2 ¼ ½a; ½a; c�c� ¼ ½a; ½ac; c��; however, ½ac; c� lies in Z as welland so ½a; c�2 ¼ 0. Consequently, ½a; c� ¼ 0. &

Corollary 2.2 therefore implies a well-known theorem of Herstein [10,Theorem 1] (see also [12, p. 4]) asserting that

(i) a simple algebra is simple as a Jordan algebra,that is, it does not contain nontrivial Jordan ideals. Another noteworthy exam-

ple of an algebra satisfying Algð½A;A�Þ ¼ A is the matrix algebra A ¼ MnðBÞ,where n5 2 and B is an arbitrary unital algebra. Indeed, in view of (2) we onlyneed to show that ½½A;A�;A� contains an invertible element, which can be easilychecked (for instance, ½½A;A�;A� contains all matrix units eij, i 6¼ j, sinceeij ¼ 1

2½½eij; eji�; eij�; the linear span of all eij, i 6¼ j, contains many invertible ele-

ments). Therefore,(ii) a Jordan ideal of MnðBÞ is an ideal.This result is not new; in fact, even a slightly more general theorem was

established by Jacobson and Rickart [13, Theorem 11] a long time ago. Anyhow,the short proof just given illustrates the usefulness of our approach, and indirectlyit also gives a new proof of the result by Fong, Miers and Sourour [6, Theorem 3].They proved that

(iii) a Jordan ideal of BðHÞ is an ideal;here, BðHÞ denotes the algebra of all bounded operators on a Hilbert space H.

However, BðHÞ is isomorphic to MnðBÞ (with B ¼ C if H is finite dimensionaland B ¼ BðHÞ if H is infinite dimensional), so that (iii) follows from (ii). Some-what more recently F€oorster and Nagy [7, Theorem 2] extended [6, Theorem 3] byproving that

(iv) a Jordan ideal of BðXÞ, where X ¼ lp, 14 p<1, or X ¼ c0, is anideal.

This theorem follows immediately from Corollary 2.2 and [7, Lemma 1] whichstates that every element in BðXÞ, where X is as above, is the sum of twocommutators. Likely there are many other Banach spaces X for which Corollary2.2 is applicable (though it can not be used for every Banach space: there exists aBanach space X such that BðXÞ has a multiplicative functional [17, 19], and soAlgð½BðXÞ;BðXÞ�Þ 6¼BðXÞ).

Jordan Ideals Revisited 3

Our next goal is to generalize the result by Fong, Miers and Sourour in anotherdirection. First we need a lemma similar to [3, Lemma 4].

Lemma 2.5. Let A be a unital C�-algebra. Then the following conditions areequivalent:

(a) Algð½A;A�Þ ¼ A;(b) Idð½A;A�Þ ¼ A;(c) there are no multiplicative functionals on A.

Proof.The implications ðaÞ¼)ðbÞ and ðbÞ¼)ðcÞ are trivial, so we only need toprove that ðcÞ¼)ðaÞ. Suppose that ðaÞ does not hold true. By (2) we then haveIdð½½A;A�;A�Þ 6¼A. Since A is unital, the norm closure I of Idð½½A;A�;A�Þ isalso a proper ideal ofA. TheC�-algebraB ¼ A=I satisfies ½½B;B�;B� ¼ 0. SinceC�-algebras do not contain nonzero nilpotent central elements, Lemma 2.4 tells usthat B is commutative. Hence there exist multiplicative linear functionals on B,which implies that there also exist multiplicative linear functionals on A. &

Corollary 2.6. A Jordan ideal of a von Neumann algebra is an ideal.

Proof. Let A be a von Neumann algebra, and let e be a central idempotent suchthat the algebra ð1 � eÞA is commutative and eA does not contain a commutativedirect summand (i.e., eA has no central portion of type I1). By [2, Lemma 2.6] eAsatisfies the condition ðbÞ of Lemma 2.5, and so it also satisfies ðaÞ, that is,Algð½eA; eA�Þ ¼ eA. Now let J be a Jordan ideal of A. Using Theorem 2.1it follows that eAJ � J. On the other hand, since ð1 � eÞA is commutative, it istrivial that ð1 � eÞAJ � J. Therefore AJ � J, i.e. J is an ideal. &

Civin and Yood [4, Theorem 5.3] proved that

(v) a closed Jordan ideal of a C�-algebra is an ideal.This result can be deduced from Corollary 2.6. The following argument was

suggested to us by Professor Bojan Magajna. Let J be a closed Jordan ideal of aC�-algebra A. We identify A with its image under the universal representationon some Hilbert space. Let A (resp. J) be the weak-operator closure of A (resp.J). Then A is a von Neumann algebra and J is its Jordan ideal, so J is actuallyan ideal of A by Corollary 2.6. Since J ¼ J \A [14, p. 713] it follows that J isan ideal of A.

There exist algebras, which can be regarded as rather nice from the structuralpoint of view, but they do contain Jordan ideals different from ideals. Let us giveone example.

Example 2.7. Let A ¼ �hx; yi be the free algebra in noncommuting variablesx and y. Let J be the �-submodule generated by x, y, x2, y2, xyþ yx and allmonomials of degree 5 3. Then J is a Jordan ideal which is not an ideal.

We remark that the Jordan ideal J in this example clearly contains nonzeroideals. In fact, another result by Herstein [12, Theorem 1.1], a generalization of (i),states that

(vi) a nonzero Jordan ideal of a semiprime algebra contains a nonzero ideal.


This can be also easily proved using our approach. Indeed, assume that A is asemiprime algebra and J is its Jordan ideal which does not contain nonzeroideals. Then ½½A;A�;A�AJ ¼ 0 by Theorem 2.1. In particular, ½½A;J�;A�A½½A;J�;A� ¼ 0 which yields ½½A;J�;A� ¼ 0 since A is semiprime. Lemma2.4 now tells us that J � Z. However, then J itself is an ideal, and so J must be0 in view of our assumption.

In order to find examples of Jordan ideals containing no nonzero ideals itseems natural to search them in algebras A satisfying ½½A;A�;A� ¼ 0. Let uspresent one such simple example.

Example 2.8. Let A be the algebra generated by x, y and relations x2 ¼ y2 ¼xyþ yx ¼ 0 (i.e., A is the nonunital Grassmann algebra in two generators). Then�x is a Jordan ideal which does not contain nonzero ideals.

3. Jordan Ideals of Symmetric Elements

Let A be an algebra with (�-linear) involution �. Following Herstein wedenote by S the Jordan algebra of all symmetric elements in A, and by K theLie algebra of all skew-symmetric elements in A. We remark that ½S;S� � K. AJordan ideal of S is of course a �-submodule J of S such that s � x2J for alls2S and x2J. An obvious example of a Jordan ideal is the set I \S where Iis a �-ideal of A (i.e., I� ¼ I), and the standard problem is to describe Jordanideals of S through those of the form I \S. In this context, our method gives thefollowing result.

Theorem 3.1. Let A be an algebra with involution �. If K ¼ ½S;S�, thenevery Jordan ideal of S is of the form I \S for some �-ideal I of A.

Proof. Let J be a Jordan ideal of S. Using (1) we see that ½½s; t�; x� 2J for alls; t2S, x2J. In view of our assumption this means that ½K;J� � J. On theother hand, S �J � J. Since A ¼ S�K these two relations together implythat

axþ ðaxÞ�; xbþ ðxbÞ� 2J for all a; b2A; x2J: ð3ÞTherefore, using

axbþ b�xa� ¼�aðxbþ ðxbÞ�Þ þ

�aðxbþ ðxbÞ�Þ

��ab�xþ ðab�xÞ�

�

we see that

axbþ ðaxbÞ� 2J for all a; b2A; x2J: ð4ÞLet I ¼ IdðJÞ. Clearly I is a �-ideal and I \S ¼ fuþ u� j u2Ig. Therefore,applying (3) and (4) it follows that J ¼ I \S. &

Unfortunately, there exist simple (even division!) algebras in which K 6¼½S;S� (and ½S;S� 6¼ 0) [15], and so Theorem 3.1 does not cover Herstein’stheorem that the simplicity of the algebra A implies the simplicity of the Jordanalgebra S [11, Theorem 8]. Anyway, there certainly exist algebras in which K ¼½S;S� holds. Let us point out a sufficient condition for this being true. Let A be


an arbitrary algebra with involution. It is easy to see, in particular by applying theJacobi identity, that Sþ ½S;S� is a Lie ideal of A. An inspection of Herstein’sarguments on Lie ideals [12, pp. 4–5] then shows that ½Idð½S;S�Þ;A� �Sþ ½S;S�, and hence K \ ½Idð½S;S�Þ;A� � ½S;S�. Therefore, K ¼½S;S� will be true if K � ½A;A� and Idð½S;S�Þ ¼ A (say, if A is unitaland some element in ½S;S� is invertible). A simple concrete example is thealgebra Mnð�Þ, n5 2, equipped with the transpose involution.

4. "-Jordan Ideals of Graded Algebras

We begin by introducing the terminology and fixing the notation. For moredetails concerning the notions that we shall define we refer the reader to the recentpapers [1, 18].

Let G be an abelian group. When considering G-graded algebras we shall, forconvenience, assume that � is a field. However, in the case of superalgebras (i.e.when G ¼ Z2), this assumption is not necessary.

Let A be a G-graded associative algebra. That is, there are subspaces Ag,g2G, of A such that A ¼ �g 2GAg and AgAh � Agh for all g; h2G. We saythat an element a2A is homogeneous if a2Ag for some g2G. The set of allhomogeneous elements in A will be denoted by H, i.e. H ¼ [g 2GAg. A sub-space V of A is said to be graded if V ¼ �g 2GV \Ag. Note that the idealIdðVÞ is graded in the case when V is a graded subspace. We say that A isgraded-semiprime if it has no nonzero nilpotent graded ideals.

Let " : G�G ! �� be a fixed anti-symmetric bicharacter. That is, " is ahomomorphism in each argument and "ðg; hÞ ¼ "ðh; gÞ�1

for all g; h2G. We shalluse the same symbol, ", to denote the map from H�H to �� definedby "ða; bÞ ¼ "ðg; hÞ where a2Ag and b2Ah. Clearly, " satisfies "ðab; cÞ ¼"ða; cÞ"ðb; cÞ and "ða; bÞ ¼ "ðb; aÞ�1

for all a; b; c2H. Consequently, "ða; aÞ ¼�1 for every a2H. If V is a graded subspace of A, then we denote by Vþ (resp.V�) the linear span of all v2V \H such that "ðv; vÞ ¼ 1 (resp. "ðv; vÞ ¼ �1).Of course, V ¼ Vþ �V�.

Introducing a new product in A by a�"b ¼ abþ "ða; bÞba (resp. ½a; b�" ¼ab� "ða; bÞba) for all a; b2H, A becomes a Jordan (resp. Lie) color algebra.The "-center of A is Z" ¼ fc2A j ½c; a�" ¼ 0 for all a2Ag. Note that Z" is agraded subspace. Finally we recall the (rather self-explanatory) definition of themain object that we are interested in: a graded subspace J of A is called an "-Jordan ideal of A if a �" x2J for all a2A, x2J.

"-Jordan ideals of graded algebras were studied by Bergen and Grzeszczuk [1],and in the case of superalgebras by G�oomez-Ambrosi and Montaner [9]. They havegeneralized Herstein’s results on Jordan ideals to the more general graded context.The proofs in these two papers are based on appropriate extensions of Herstein’sclassical construction. Adapting the arguments from Section 2 we shall be able toconsider "-Jordan ideals in a different way. Our clue is the identity

½½a; b�"; c�" ¼ a �" ðb �" cÞ � "ða; bÞb �" ða �" cÞ; ð5Þwhich one can easily check. Here, a; b; c are arbitrary elements in H; if"ða; bÞ ¼ "ða; cÞ ¼ "ðb; cÞ ¼ 1, then (5) coincides with (1).


Theorem 4.1. If J is an "-Jordan ideal of a graded algebra A, then Jcontains the set Algð½A;A�"ÞJ. In particular, J contains the graded idealIdð½½A;A�";A�"ÞJ.

Proof. The proof is just a modification of that of Theorem 2.1, so we just sketchit. First use (5) to conclude that ½½a; b�"; x�"2J for all a; b2A, x2J, from whichAlgð½A;A�"ÞJ � J follows. Next, apply

a½½b; c�"; d�" ¼ ½a½b; c�"; d�" � "ðbc; dÞ½a; d�"½b; c�"to derive that

Idð½½A;A�";A�"Þ � Algð½A;A�"Þ: ð6ÞFinally, note that Idð½½A;A�";A�"ÞJ is a graded ideal. &

Accordingly, Corollaries 2.2 and 2.3 can be extended as follows.

Corollary 4.2. Let A be such that Algð½A;A�"Þ ¼ A. Then every "-Jordanideal of A is an ideal.

Corollary 4.3. Let A be such that Idð½½A;A�";A�"Þ has either trivial rightannihilator or trivial left annihilator. Then every nonzero "-Jordan ideal of Acontains a nonzero graded ideal.

We continue by extending Lemma 2.4.

Lemma 4.4. Let A be such that Z" does not contain nonzero nilpotent ele-ments. If c2Aþ satisfies ½½A; c�";A�" ¼ 0, then c2Z".

Proof. We have ½A; c�" � Z". We may assume that c2H. Let a2H. Us-ing ½a; c�"2Z" we get ½a; c�2" ¼ "ðac; aÞ½a; ½a; c�"c�". Since "ðc; cÞ ¼ 1 we have

½a; c�"c ¼ ½ac; c�"2Z" and so it follows that ½a; c�2" ¼ 0. According to our assump-tion this implies ½a; c�" ¼ 0. &

Corollary 4.5. Let A be graded-semiprime. If J is an "-Jordan ideal of Awhich does not contain a nonzero graded ideal of A, then Jþ ¼ 0 and the idealI ¼ IdðJÞ satisfies Iþ � Z", ½I�;Aþ�" ¼ 0 and I� �" A� ¼ 0.

Proof. Theorem 4.1 tells us that Idð½½A;A�";A�"ÞJ ¼ 0. Note that this im-plies that Idð½½A;A�";A�"ÞI ¼ 0 as well. Consequently, Idð½½A;I�";A�"Þ

2 ¼ 0,which in turn yields Idð½½A;I�";A�"Þ ¼ 0 since A is graded-semiprime. Thus,½½A;I�";A�" ¼ 0. It is clear that the "-center of a graded semiprime algebradoes not contain nonzero nilpotents. Therefore we infer from Lemma 4.4 thatIþ � Z". In particular, Jþ � Z". Thus, for x2Jþ and a2A we havexa ¼ 1

2x �" a2J. That is, J contains the graded ideal IdðJþÞ. But then Jþ ¼ 0

in view of our assumption. Further, as a special case of ½½A;I�";A�" ¼ 0 we have½I�;Aþ�" � ðZ"Þ�. However, it is very easy to see that ðZ"Þ� ¼ 0 [18, Lemma1.3] (indeed, just note that if c2ðZ"Þ� then c2 ¼ 1

2½c; c�" ¼ 0). Therefore,

½I�;Aþ�" ¼ 0. Finally, pick y2I� \H and b2A� \H. Then by 2Iþ � Z"

and so ðbyÞb ¼ "ðby; bÞb2y ¼ �"ðy; bÞb2y; that is, bðy �" bÞ ¼ 0. Sincey �" b2Iþ � Z" it follows that ðy �" bÞ2 ¼ 0, and hence y �" b ¼ 0 since A isgraded-semiprime. Thus, I� �" A� ¼ 0. &


Corollary 4.6. Let A be graded-semiprime. Then the following conditions areequivalent:

(a) there exists an "-Jordan ideal ofA which does not contain nonzero gradedideals of A;

(b) there exists a graded ideal I of A such that Iþ � Z", I� 6¼ 0,½I�;Aþ�" ¼ 0 and I� �" A� ¼ 0.

Proof. The implication ðaÞ¼)ðbÞ follows from Corollary 4.5. Assume that ðbÞholds. Using I� �" A� ¼ 0 we see that I� is an "-Jordan ideal of A. If I�contains a subalgebra U, then U2 � Iþ \U � Iþ \I� ¼ 0. Therefore, I�cannot contain nonzero graded ideals of A. &

Corollaries 4.5 and 4.6 sharpen [1, Theorem 2.3]. In the case of graded simplealgebras, however, Bergen and Grzeszczuk gave a more profound description ofthe situation treated in these two corollaries (see [1, Proposition 2.7, Theorem2.8]).

A Z2-graded associative algebra A is called an associative superalgebra; thus,A ¼ A0 �A1 and AiAj � Aiþj, i; j2Z2. In this context we consider thebicharacter, now denoted by s to distinguish it from the one treated in the generalcase, defined by sð1; 1Þ ¼ �1 (and of course sði; jÞ ¼ 1 if one of i; j is 0). Accord-ingly, Aþ ¼ A0 and A� ¼ A1. Note that the three conditions Iþ � Z",½I�;Aþ�" ¼ 0 and I� �" A� ¼ 0, can be in this setting simply read asI � Z (the ordinary center of A). Therefore, Corollary 4.5 implies

Corollary 4.7. Let A be a semiprime associative superalgebra. If J is an s-Jordan ideal of A which does not contain nonzero graded ideals of A, thenJ0 ¼ 0 and IdðJÞ � Z.

It is easy to see that a prime associative superalgebra must be commutative if itcontains a nonzero graded ideal that lies in the center. Therefore Corollary 4.7generalizes [9, The first Herstein construction for superalgebras].

We conclude this section by pointing out a condition under which s-Jordanideals are necessarily Jordan ideals.

Corollary 4.8. Let A be a unital associative superalgebra. Suppose that everyautomorphism of the algebra A is inner. If J is an s-Jordan ideal of A, thenA1J � J; in particular, J is a Jordan ideal of A.

Proof. Since a0 þ a1 7! a0 � a1, ai2Ai, is an automorphism, there existsan invertible t2A such that a0 � a1 ¼ tða0 þ a1Þt�1 for all ai2Ai; that is,tai ¼ ð�1Þiait. If t ¼ t0 þ t1 then t0 � t1 ¼ ttt�1 ¼ t which implies t1 ¼ 0, i.e.t2A0. Since each a1 2A1 can be written as a1 ¼ 1

4t�2½½a1; t�s; t�s it follows from

Theorem 4.1 that a1J � J. &

It is well-known that, for every Banach space X, the algebra BðXÞ has onlyinner automorphisms [5]. Thus, whichever Z2-grading on BðXÞ we take, s-Jordanideals of A are just Jordan ideals. Moreover, if X is a Hilbert space, X ¼ lp,14 p<1, or X ¼ c0, then s-Jordan ideals of BðXÞ are necessarily ideals (seeSection 2).


5. "-Jordan Ideals of Symmetric Elements

We keep the notation from the previous section. So, letA be aG-graded algebra.A linear map � : A ! A is a color involution if A�

g ¼ Ag for all g2G, a�� ¼ afor all a2A, and ðabÞ� ¼ "ða; bÞb�a� for all a; b2H. As in the non-gradedcase we define that a2A is symmetric (resp. skew-symmetric) if a� ¼ a (resp.a� ¼ �a). The set S (resp. K) of all symmetric (resp. skew-symmetric) elementsin A is a Jordan (resp. Lie) color algebra. Note that ½S;S�" � K. By an "-Jordanideal ofS we of course mean a graded subspace J of S such that s �" x2J for alls2S, x2J. Herstein’s results on Jordan ideals of symmetric elements wereextended to "-Jordan ideals in [1], and in the case of superalgebras in [8, 9]. Weconclude this paper by extending Theorem 3.1 to "-Jordan ideals.

Theorem 5.1. Let A be a graded algebra with color involution �. IfK ¼ ½S;S�", then every "-Jordan superideal of S is of the form I \S forsome graded �-ideal I of A.

Proof. The proof is a straightforward extension of that of Theorem 3.1, so wejust sketch it. Let J be a Jordan superideal of S. Applying (5) and our assumptionthat K ¼ ½S;S�" we obtain ½K;J�" � J, which together with S �" J � Jgives axþ ðaxÞ� ¼ axþ "ða; xÞxa� 2J and xbþ ðxbÞ� ¼ xbþ "ðx; bÞb�x2Jfor all a; b2H, x2J \H. Therefore, using

axbþ "ða; bÞ"ðx; bÞ"ða; xÞb�xa�

¼�a�xbþ "ðx; bÞb�x

�þ "ða; xbÞ

�xbþ "ðx; bÞb�x

�a�

�

� "ðx; bÞ�ðab�Þxþ "ðab; xÞxðab�Þ�

�

we see that axbþ ðaxbÞ� 2J for all a; b2H, x2J \H. Now it is easy to checkthat I ¼ IdðJÞ is a graded �-ideal and J ¼ I \S. &

Acknowledgement. The authors would like to thank the referee for careful reading of the paper anduseful suggestions.

References

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[2] Bre�ssar M (1991) Centralizing mappings on von Neumann algebras. Proc Amer Math Soc 111:501–510

[3] Bre�ssar M (1992) Characterizations of derivations on some normed algebras with involution.J Algebra 152: 454–462

[4] Civin P, Yood B (1965) Lie and Jordan structures in Banach algebras. Pacific J Math 15: 775–797[5] Eidelheit M (1940) On isomorphisms of rings of linear operators. Studia Math 9: 97–105[6] Fong CK, Miers CR, Sourour AR (1982) Lie and Jordan ideals of operators on Hilbert space. Proc

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ciative superalgebras. Comm Algebra 28: 3743–3762


[10] Herstein IN (1955) On the Lie and Jordan rings of a simple, associative ring. Amer J Math 77:279–285

[11] Herstein IN (1956) Lie and Jordan systems in simple rings with involution. Amer J Math 78:629–649

[12] Herstein IN (1969) Topics in Ring Theory. Chicago: The University of Chicago Press[13] Jacobson N, Rickart C (1950) Jordan homomorphisms of rings. Trans Amer Math Soc 69:

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J Algebra 195: 558–579[19] Wilansky A (1971) Subalgebras of BðXÞ. Proc Amer Math Soc 29: 355–360

Authors’ addresses: M. Bre�ssar (e-mail: [email protected]) and A. Fo�ssner (e-mail: [email protected]), Department of Mathematics, University of Maribor, PEF, Koro�sska 160, 2000 Maribor,Slovenia; M. Fo�ssner, Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana,Slovenia, e-mail: [email protected]

10 M. Bre�ssar et al.: Jordan Ideals Revisited

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Jordan Ideals Revisited