On the derivations of the quadratic Jordan product in classical finite dimensional Cartan factors

Journal of Algebra 635 (2023) 411–435 Contents lists available at ScienceDirect Journal of Algebra journal homepage: www.elsevier.com/locate/jalgebra On the derivations of the quadratic Jordan product in classical finite dimensional Cartan factors José M. Isidro Santiago de Compostela, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 15 December 2022 Available online 18 August 2023 Communicated by Alberto Elduque MSC: 47B49 46K15 Keywords: JB∗-triples Classical Cartan factors JB∗-triple derivations Banach-Lie groups Let Rbe a unital *-ring which contains a complex unit and in which the unit can be halved. Conmutativity of Ris not assumed. Let Mnand Snbe the Jordan *-triples of the square n ×nmatrices (respectively, n ×nsymmetric matrices) (n ≥2) with entries in R. The additive groups Der (Mn)and Der (Sn) of the quadratic Jordan derivations of Mn(respectively, Sn) are described in terms of Der (R), the group of quadratic Jordan derivations of R. Particular attention is paid to the case R =C. © 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). 1. Introduction and preliminaries Let Rbe a unital *-ring, that is, a ring with a unit 1and an involution a → a∗ that satisfies (a∗)∗=a, (a +b)∗=a∗+b∗, (ab)∗=b∗a∗and 1∗=1for a, b ∈R. Commutativity of Ris not assumed. Besides we endow Rwith the Jordan product defined by a ◦b := 1 2(ab +ba) (this requires that the unit, hence any element in R, can be halved, i.e., for a ∈Rthere is b ∈Rwith 2b =a) and with the Lie product given by a, b:= ab −ba. A ring derivation is a map φ:R →Rsuch that φ(ab) =φ(a)b +aφ(b) E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2023.08.004 0021-8693/© 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). 412 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 holds for a, b ∈R. A derivation δof Ris said to be a *-derivation [respectively, a skew *-derivation, an additive derivation] if δ(a∗) =δ(a)∗[respectively, δ(a)∗=−δ(a∗), δ(a +b) =δ(a) +δ(b)] holds for a, b ∈R. Some examples: if ais a skew symmetric element of R[that is, if a∗=−a] then the map Ja(x) := a ◦x, (x ∈R) of Jordan multiplication by aand the map Ad (a)x := a, x, x ∈R, of Lie multiplication by aare, respectively, an additive skew *-derivation and an additive *-derivation of R. We set der(R)for the set of the *-derivations of the *-ring R. We also consider the quadratic Jordan product and the Jordan triple product in R defined respectively by {aba}:= ab∗a, and {abc}:= 1 2(ab∗c+cb∗a)(a, b, c ∈R). A quadratic (respectively, a triple) Jordan derivation of Ris a map δ:R →Rthat satisfies the corresponding below functional equation δ{aba}={δ(a)ba}+{aδ(b)a}+{abδ(a)}(a, b ∈R) δ{abc}={δ(a)bc}+{aδ(b)c}+{abδ(c)}(a, b, c ∈R), and we set Der (R) (respectively, Der(R)) for the set of them. We remark that in the above definition of δthe quadratic and the triple Jordan product are involved, but no reference is made to the behavior of δrelative to the * operation in R. Remark that in general neither der(R) ⊂Der (R)nor Der (R) ⊂der(R), however additive elements of der(R)belong to Der (R)[see Example 2.4 below]. Notice also that Der(R) ⊂Der (R) and that additive elements in Der (R)belong to Der(R). The involution ∗of Rinduces a direct sum decomposition R =R+⊕R −where R+:= a∈R:a=a∗and R−:= a∈R:a∗=−a.(1) Here R+and R−are not subrings of R, however R+is closed under the Jordan product, and both R+and R−are closed under the quadratic and the triple Jordan product of Rand 1 ∈R +. The ring, the Jordan and the Lie operations in Rextend in a natural way to the set Mn(n ≥2) of square n ×nmatrices with entries in Robtaining another unital ring. For A =(aij) ∈Mnwe have three candidates for the *-operation: the transposition A → Atwhich interchanges rows with columns in A, the conjugation A → A†:= (a∗ ij) which extends the * operation of R, and the adjunction A → A∗:= (At)†=(A†)twhich combines them. Note that we use the same symbol ∗to denote the involution in Rand the adjunction in Mn, but there is no danger of confusion since the elements of Mnare denoted with capital characters. The quadratic and the triple Jordan product in Mnare given respectively by {ABA}:= AB∗Aand {ABC}:= 1 2(AB∗C+CB∗A)(A, B, C ∈Mn).(2) J.M. Isidro / Journal of Algebra 635 (2023) 411–435 413 The quadratic (respectively, the triple) Jordan derivations of Mnare the maps δ:Mn→ Mnthat satisfy the corresponding below functional equation δ{ABA}={δ(A)BA}+{Aδ(B)A}+{ABδ(A)}(A, B ∈Mn)(3) δ{ABC}={δ(A)BC}+{Aδ(B)C}+{ABδ(C)}(A, B, C ∈Mn).(4) In analogy to (1), each one of the involutions σ∈{t, †, ∗} of Mngives rise to a direct sum decomposition Mn=Sσ n⊕Aσ nif we define Sσ n:= A∈Mn:Aσ=Aand Aσ n:= A∈Mn:Aσ=−A. Here neither Sσ nnor Aσ nis closed under the usual multiplication of matrices, however both of them are closed under the quadratic and the triple Jordan product, hence they are Jordan subtriples of Mn. Moreover Sσ nis closed under the Jordan product and 1∈Sσ n. The elements in St n(respectively At n) are said to be symmetric (respectively skew- or antisymmetric); the elements in S† n(respectively A† n) are said to be real (respectively imaginary), and those in S∗ n(respectively A∗ n) are referred to as hermitian and skewhermitian elements respectively. The sets Der (Mn), Der (Sσ n)and Der (Aσ n)of the corresponding quadratic Jordan derivations [and their respective triple Jordan versions Der(Mn), Der(Sσ n)and Der(Aσ n))] are clearly additive groups. In this paper we only consider the transposition (the involution t, hence in the sequel we omit any reference to σ), and our purpose is to identify the elements in Der (Mn)and Der (Sn)[and their respective triple Jordan versions Der(Mn) and Der(Sn(Mn)] in terms of those in Der (R). Throughout the whole paper we assume that the unit 1 ∈Rcan be halved and that Rcontains a complex unit, that is, an element χ ∈Rsuch that χ2=−1and χ∗=−χ, conditions that are clearly satisfied if R =Cor R =H(the ring of quaternions). Some examples: if A∗=−Ais a skew-hermitian element in Mnthen the map JA(X) := A ◦X, (X∈Mn), of Jordan multiplication by A ∈Mn, and the map Ad (A)X:= A, X, (X∈Mn), of Lie multiplication by Aare, respectively, an additive skew *-derivation and an additive *-derivation of Mn. And due to the matrix structure of Mn, a new type of derivation comes into scene: let φ ∈der(R)be an additive *-derivation of R, then φ extends to an additive *-derivation of Mnby setting X→ Xφ:= φ(xij)(X=(xij)∈Mn).(5) In the second section of this article we solve the above mentioned problem for the matrix ring Mn, n ≥2. We let 1denote the unit matrix in Mnand prove that any δ∈Der (Mn)with n ≥2admits a representation of the form δ(X)=Jδ(1)(X)+Ad (T)X+Xφ(X∈Mn),(6) 414 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 where δ(1)∗=−δ(1)and T∗=−Tare skew-hermitian elements of Mnand φ, which does not depend on n, is an additive *-derivation on Rwith φ(1) =0. In particular, for n ≥2all elements in Der (Mn)are additive and hence Der (Mn) =Der(Mn). In the third section of the paper we consider Sn. We do not know whether or not all elements in Der (Sn)are additive and prove that all additive elements in Der (Sn)admit the expression (6), though the operator Tnow satisfies T∗=Tt=−T. So far our considerations remain in a purely algebraic setting and the arguments and calculations we make involve only the mathematical structure of a *-ring. In the fourth section we assume Rto be endowed with a topology τsuch that (R, τ)is a locally compact topological *-ring in which the * operation is continuous. For each fixed n ∈N, we equipped Rnwith the product topology τn, hence each matrix A ∈Mncan be viewed as a continuous function fA:Mn→Mngiven by the matrix product fA(x):=Axwhere x=(x1,··· ,x n)∈R n. There are several candidates for a topology on Mn, and we consider the compact-open topology: it makes Mninto a topological *-ring in which any of the candidates for the *-operation (i.e., the transposition t, or the conjugation †, or the adjunction ∗) are continuous. Any map δ∈Der (Mn) [respectively, any additive δ∈Der (Sn)] admits the representation (6) and, since the maps Jδ(1)and Ad (T) automatically are additive and continuous on the *-ring Mn, a derivation δ∈Der (Mn) [respectively, any additive δ∈Der (Sn)] is continuous if and only if so is the map A → Aφ, A ∈Mn, which in turn occurs if and only if φ ∈Der (R)is continuous. Of particular interest is the case in which the basic *-ring Ris Cand the *-operation in Ris the complex conjugation. In [[5], Lemma 2.3] it was proven that each additive element φ ∈Der (C)is the multiplication by φ(1), hence the only additive element φ in Der (C)such that φ(1) =0is the null map, which gives a simpler expression for the quadratic derivations of Mn(C)and for those of Sn(C)that are additive since in (6)now the term Xφvanishes. Recall that Mn(C)and Sn(C)are respectively the classical finite dimensional Cartan factors of types Inand IIn. Next we discuss the quadratic additive Jordan derivations of type IVnCartan factors. These are known to be JB∗-triple isomorphic to special JB∗-subalgebras X ⊂Sn(C)and the arguments and calculations that we made for the additive elements of Der (S2(C)) can be implemented in the subalgebra Xwhere they remain valid. Notice that in the definition of quadratic Jordan derivations neither the linearity nor the continuity of δwas assumed, however as a consequence of the explicit formula (6)it turns out that quadratic [in its case, additive quadratic] Jordan derivations of these finite dimensional classical Cartan factors are automatically complex linear and continuous. Our general references for background on JB∗-triples, in particular for Cartan factors, are [7], [8]and [1]. For Jordan algebras we refer to [2]. J.M. Isidro / Journal of Algebra 635 (2023) 411–435 415 2. Quadratic Jordan derivations of Mn Below we prove the following 2.1 Theorem. Let Rbe a unital *-ring in which there is a complex unity and the unit can be halved, and let Mn, n ≥2, be the ring of all square n ×nmatrices with entries in R. Then each element Φ ∈Der (Mn)is of the form Φ(A)=JΦ(1)(A)+Ad (T)A+Aφ(A∈Mn)(7) for a suitable matrix T∈Mnwith T∗=−Tand some additive φ ∈der(R)which does not depend on n and satisfies φ(1) =0. The representation is unique up to a term of the form Ad (T0)for some T0in the center Z(Mn)with T∗ 0=−T0. All elements in Der (Mn) are additive. The proof will consist of an induction argument on nfor which we need some preliminary results. Some examples and general results The below examples of quadratic Jordan derivations in Mnare decisive in our study. 2.2 Example. Let X∈Mnbe such that X∗=−X. Then the map JX:Mn→Mngiven by JX(A) := X◦Afor A ∈Mnis an additive skew *-derivation of Mn. Proof. We have to check that JX{ABA}={JX(A)BA}+{AJX(B)A}+{ABJX(A)} holds for all A, B∈Mnwhich, by the assumption X∗=−X, reads (AB∗A)X+X(AB∗A)=(AX +XA)B∗A+AB∗(AX +XA)+ABX +XB∗A =(AX +XA)B∗A+AB∗(AX +XA)+AX∗B∗+B∗X∗)A =(AX +XA)B∗A+AB∗(AX +XA)−AXB∗+B∗X)A which is easily checked to be true. Moreover JX(A)∗=JX∗(A∗) =−JX(A∗).  Also decisive are the two following examples, proofs for which can be found in [5] Examples 2.1 and 2.2 where they were studied in the broader setting Mn,m of the rectangular n ×mmatrices. 2.3 Example. Let U∈Mnbe such that U∗=−Uand define a map Ad (U)A := U, A for A ∈Mn. Then Ad (U)is an additive *-derivation of Mn. 416 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 2.4 Example. To each A =(aij) ∈Mnand each additive *-derivation φof Rwe define Aφ:= φ(aij)∈Mn. Then the map A → Aφis a quadratic additive *-derivation of Mn. 2.5 Remark. The derivations δ∈Der (Mn)in the above examples (2.2), (2.3)and (2.4) are all additive, hence in fact they belong to smaller set Der(Mn). We shall frequently use the following form of condition (3)in which Jordan triple products {A BC}are replaced with the usual products ABC. 2.6 Lemma. If δ∈Der (Mn)commutes with the * -operation in Mni.e., if δ(A∗) =δ(A)∗ holds for A ∈Mn, then condition (3)is equivalent to δ(ABA)=δ(A)BA +Aδ(B)A+ABδ(A),(A, B ∈Mn).(8) Proof. Quite elementary.  2.7 Lemma. Let n ≥2be any fixed positive integer and let δ∈Der (Mn). Then δ(0) =0 and δ(1) =−δ(1)∗. Proof. From 0 ={0 10}and (3)we get δ(0) =δ{0 10} =2{δ(0) 10} +{0 δ(1) 0} =0. Similarly from 1={111}and (3)we get δ(1)=δ{111}=2{δ(1)11}+{1δ(1)1} =2δ(1)1∗1+1δ(1)∗1=2δ(1)+δ(1)∗. 2.8 Lemma. Let n ≥2be fixed and let δ∈Der (Mn). Define a map D:Mn→Mnby D:= δ−Jδ(1)A. Then D∈Der (Mn)and D(1)=0,(9) D(A2)=A◦D(A)+D(A)◦A, (10) D(A∗)=D(A)∗.(11) In particular, for Dwe can use the form (8)of equation (3). Proof. We have δ(1)∗=−δ(1)by Lemma 2.7 hence, by Example 2.2, the operator Jδ(1)of Jordan multiplication by δ(1)is additive and Jδ(1)∈Der (Mn), whence so is D:= δ−Jδ(1). Clearly D(1) =0. From A2={A 1A}and (3) applied to Dwe get D(A2)=D{A1A}=2{D(A)1A}+{AD(1)A} =2 D(A)◦1∗◦A−1∗◦A◦D(A)+A◦D(A)◦1∗ =A◦D(A)+D(A)◦A. J.M. Isidro / Journal of Algebra 635 (2023) 411–435 417 From A∗={1A 1}and (3) applied to Dwe get D(A∗)=D{1A1}=2{D(1)A1}+{1D(A)1} =1◦D(A)∗◦1−D(A)∗◦1◦1+1◦1◦D(A)∗=D(A)∗. Quadratic Jordan derivations of M2 Since we are going to make an induction argument on n ≥2, we begin with the study of Der (M2). The elements of M2are square 2 ×2 matrices A =(aij)with entries aij ∈R. Of particular relevance for the proof will be the matrices E11 := 10 00 E12 := 01 00 E21 := 00 10 E22 := 00 01 . Clearly E11 and E22 are pairwise orthogonal projections in M2with E11 +E22 =1 whereas E12 and E21 are tripotents in M2with E∗ 12 =E21. 2.9 Lemma. Let δ∈Der (M2). Then there is a matrix T∈M2with T∗=−Tsuch that the derivation Δ := δ−Jδ(1)+Ad (T)enjoys properties (9)to (11)and besides Δ(Eij) =0for i, j=1, 2. Proof. We let D:M2→M2be the map given by D:= δ−Jδ(1). By Lemma 2.8 and Examples 2.2 and 2.3, Dis a quadratic Jordan derivation that satisfies D(1) =0and enjoys properties (9)to (11). We let D(E11)=(aij),D(E12)=(bij),D(E21)=(cij)D(E22)=(dij) be the images of the Eij by D. Since for i =1, 2the elements Eii are projections, by (10)and (11)we have D(E∗ ii) =D(Eii)∗and D(Eii) =EiiD(Eii) +D(Eii)Eii which in terms of D(E11) =(aij)and D(E22) =(dij)gives D(E11)=0x x0and D(E22)=0y y0 for some elements x, y∈Rwith x =x∗and y=y∗. Similarly, since Dsatisfies the equation (3), the relations E∗ 12 =E21 and E12 ={E12 E12 E12}give D(E∗ 12) =D(E12)∗ and D(E12) =2{D(E12) E12 E12} +{E12 D(E12) E12}, which in terms of D(E12) =(bij) means D(E12)=z0 0w for some z, w∈Rwith z=z∗and w=w∗. From {E12 E11 E12} =0and {E12 E22 E12} = 0and (3)one gets z=−xand z=y, w=−xand we finally get 418 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 D(E11)=0x x0D(E12)=x0 0−x D(E21)=−x0 0xD(E22)=0−x −x0 for some x ∈Rwith x∗=x. Let us define T∈M2by T:= 0x −x0, hence T∗=−T. By Example 2.3 the mapping Ad (T): M2→M2is a quadratic Jordan derivation in M2 that satisfies Ad (T)1=0, hence the map Δ := D+Ad (T)satisfies Δ ∈Der (M2)and enjoys properties (9), (10), (11)as well as Δ(Eij) =0for i, j=1, 2.  Next we shall find an explicit expression for Δ. 2.10 Lemma. Let Δbe a quadratic Jordan derivation of M2that enjoys properties (9) to (11)and Δ(Eij) =0for all i, j∈{1, 2}. Then there is an additive *-ring derivation φ ∈der(R)such that φ(1) =0and Δ(A) =Aφholds for all A ∈M2. Proof. Since Δsatisfies the functional equation (3)and Δ(Eij) =0we have Δ{EijAEij}=2{Δ(Eij)AE ij}+{Eij Δ(A)Eij}={Eij Δ(A)Eij}(A∈M2).(12) For each matrix X=( xrs )∈M2and each (i, j), 1 ≤i, j≤2, we define the matrix  Xij by  Xij := EijX∗Eij ={Eij XE ij}=x∗ jiEij.(13) It is worthwhile to notice that all entries in  Xij are null except for the (i, j)-th entry which is precisely x∗ ji, that is, the *-image in Rof the (j, i)-th entry in X. Now, let A =(ars) ∈M2be arbitrary and let B:= Δ(A) =(bij)be its image by Δ. From (12) and the above we have  Bij =EijB∗Eij =EijΔ(A)∗Eij ={Eij Δ(A)Eij}=Δ{Eij AE ij}=Δ x∗ jiEij) which means that the (ij)-the entry in B=Δ(A) depends only on the (ji)-th entry in A. Therefore we may write Δ(A)=φ11(a11)φ12(a12) φ21(a21)φ22(a22)(14) for some maps φij :R →R. Furthermore, from Δ(Eij) =0for all i, j∈{1, 2}we conclude that φij(0) =0and φij(1) =0. Let J:= ΣEij ∈M2and observe that then Δ(J) =φij(1)=(0). Therefore since Δsatisfies (3)and Δ(J) =(0)we get Δ{JXJ}=2{Δ(J)XJ}+{JΔ(X)J}={JΔ(X)J}(15) J.M. Isidro / Journal of Algebra 635 (2023) 411–435 419 for all X∈M2. Now, fix any a ∈R. One directly verifies that JE11J=Jand J(aE11)J=aJ. Applying the above relations [with areplaced with φ11(a)] and using (15)we get φ11(a)J=Jφ11(a)E11)J=JΔ(aE11)J =Δ(JaE11J)=Δ(aJ)=φ11(a)φ12(a) φ21(a)φ22(a), therefore φ11 =φ12 =φ21 =φ22. We label this common map with φ, hence φ(1) =0 and Δ(A)=Δ (aij)=φ(aij) holds for all A =(aij) ∈M2. For any a, b ∈Rlet A ∈M2be given by A:= ab b0hence Δ(A)=φ(a)φ(b) φ(b)0  therefore by (14) A2=a2+b2ab ba b2and Δ(A2)=φ(a2+b2)φ(ab) φ(ba)φ(b2). On the other hand, AΔ(A)+Δ(A)A=aφ(a)+φ(a)a+bφ(b)+φ(b)baφ(b)+φ(a)b bφ(a)+φ(b)abφ(b)+φ(b)b therefore by comparing these two expressions of Δ(A2)we get φ(ab) =aφ(b) +φ(a)b. Next we show that φis additive on R. Let a, bbe arbitrary in Rand consider the matrix B:= ab b0hence Δ(B)=φ(a)φ(b) φ(b)0 . By direct calculation with J:= 11 10 we get JΔ(B)J=φ(a)+2φ(b)φ(a)+φ(b) φ(a)+φ(b)φ(a) and JBJ =a+2ba+b a+ba . 426 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 Clearly W∗ j=−Wjand Ad (Wj)Dj=−Δ(Dj)and Ad (Wj)Dk=0for k=j. This completes the proof of the claim in Step 3.  Step 4By setting W:= Σm 1Wjand replacing Δwith Ξ := Δ +Ad (W), that is, Ξ(X):=Δ(X)+Ad (W)X(X∈Mm+1) (28) we get a map Ξ ∈Der (Mm+1)that enjoys the properties of Δand besides satisfies Ξ(Dk) =0for k∈{1, 2, ··· , m}. Below in step 5 we prove that Ξ(A)=Aφ(A∈Mm+1) holds, where φis given by (23). By retrieving the definitions of Δ, Θ, Ψ, Φwe get Ξ=Δ+Ad (W)=(Θ−Ad (V)) + Ad (W) =(Ψ−Ad (U)) −Ad (V)+Ad (W)=(Φ−JΦ(1))−Ad (T) where T:= U−V+Wand therefore Φ=JΦ(1) +Ad (T)+Ξ which will complete the proof of Theorem 2.1. Step 5Claim The map Ξ ∈Der (Mm+1) constructed in (28)satisfies Ξ(A) =Aφfor all A ∈Mm+1 with φas in (23), in particular φ ∈der(R)does not depend on n, is additive and satisfies φ(1) =0. Proof. Let us fix j∈{1, 2, ··· , m}. Since by assumption m ≥2, there exists k∈ {1, 2, ··· , m}such that k=jand therefore Ej,(m+1) =DkEjkDkand E(m+1),j =DkEkjDk. Let a ∈R. We have 1 ≤j, k≤mwith j=k, hence Ejk ∈Mmand, by the induction hypothesis now applied to  Ξ:=Ξ|Mm∈Der (Mm)we get Ξ(aEjk)= Ξ(aEjk)=φ(a)Ejk whence ΞaEj,(m+1)=Ξ Dk(aEjk)Dk=DkΞφ(a)Ejk)Dk =φ(a)DkEjkDk=φ(a)Ej,(m+1),(29) ΞaE(m+1),j=Ξ Dk(aEkj)Dk=DkΞφ(a)Ekj)Dk =φ(a)DkEkjDk=φ(a)E(m+1),j .(30) J.M. Isidro / Journal of Algebra 635 (2023) 411–435 427 Moreover E(m+1)(m+1) =D1E11D1whence ΞaE(m+1)(m+1)=Ξ D1(aE11)D1=D1Ξ(aE11)D1 =D1φ(a)E11D1=φ(a)E(m+1)(m+1).(31) The equations (29), (30)and (31) prove that Ξ(aEjk) =φ(a)Ejk for j, k∈1, 2, ··· , m; m +1 and a ∈R. Finally, let Ξ(A) =(bjk)for A =(ajk) ∈Mm+1. Then bjkEjk =EkjΞ(A)Ejj =Ξ(EjkAEjk)=Ξ(akjEkj) which shows that Ξ(A) =φ(ajk)=Aφholds for all A ∈Mm+1. Thus Φ(A)=A◦Φ(1) + Ad (T)A+Aφ,(A∈Mm+1) where T=U+V+W∈Mm+1 satisfies T∗=−Tand φ ∈der(R)is additive, does not depend on mand satisfies φ(1) =0. The assertion concerning the uniqueness of the above representation is clear.  3. Triple Jordan derivations in Sn In this section our task will be the proof of the following 3.1 Theorem. Let Rbe a unital *-ring that contains a complex unit and in which every element can be halved, and let Snbe the Jordan *-triple of all n ×n(n ≥2) symmetric matrices with entries in R. Then each additive Φ ∈Der (Sn)is of the form Φ(A)=JΦ(1)A+Ad (T)A+Aφ(A∈Sn) (32) for a suitable matrix T∈Mnwith T∗=Tt=−Tand some additive φ ∈der(R)which does not depend on n and satisfies φ(1) =0. The representation is unique up to a term of the form Ad (T0)for some T0∈Mnthat commutes with all matrices in Sn. First we consider the Jordan triple S2. Most of the arguments that follow are a repetition of the ones we made for the ring Mnand we shall only give details when essential changes are needed. Triple Jordan derivations of S2 Our purpose is to study the group Der (S2). The elements of S2are square 2 ×2 symmetric matrices A =(aij)with entries aij ∈R, or in details A =a11 a12 a21 a22 where a12 =a21, and we set a1/2:= a12 =a21. Of particular relevance will be the matrices 428 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 E11 := 10 00 E1/2:= 01 10 E22 := 00 01 . Clearly E11 and E22 are pairwise orthogonal projections in S2with E11 +E22 =1 whereas E1/2is a tripotent in S2with E∗ 1/2=E1/2. Moreover {E1/2E11 E1/2} =E22 and {E1/2E22 E1/2} =E11. We introduce some notation to shorten the writing. For A ∈S2we define the operators Q(Eii)and Q(E1/2)from S2→S2by Q(Eii)A:= {Eii A∗Eii}=EiiAEii (i=1,2) Q(E1/2):=Id S2−Q(E11)+Q(E22)(A∈S2)(33) or in details, as an elementary matrix calculus shows, Q(Eii)A=aiiEii Q(E1/2)A=0a1/2 a1/20=a1/2E1/2(i=1,2).(34) From the definition of Q(E1/2)and A={1A∗1}={E11 +E22 A∗E11 +E22} =Q(E11)A+{E11A∗E11}+{E22A∗E11}+Q(E22)A(A∈S2) we get Q(E1/2)A =2{E11A∗E22}and hence in general Q(E1/2)A ={E1/2A∗E1/2}. 3.2 Lemma. Let δ∈Der (S2). Then there is a matrix T∈M2with T∗=Tt=−Tsuch that the derivation Δ := δ−Jδ(1) +Ad (T)enjoys properties (9)to (11)and D(Eii) =0 for i =1, 2. In particular Δcommutes with Q(Eii)for i =1, 2. Proof. Set D:= δ−Jδ(1). Since 1∈S2and S2is closed under the Jordan product, by Examples 2.2 and 2.3 we get that D∈Der (S2)and clearly it satisfies D(1) =0 and enjoys properties (9)to (11). Since for i =1, 2the elements Eii are projections, by (10)and (11)we have D(E∗ ii) =D(Eii)∗and D(Eii) =EiiD(Eii) +D(Eii)Eii and the argument in the proof of Lemma 2.9 shows that D(E11)=0x x0and D(E22)=0y y0 for some elements x, y∈Rwith x =x∗and y=y∗. Similarly, since E1/2is a tripotent and Dsatisfies (3), the relations E∗ 1/2=E1/2and E1/2={E1/2E1/2E1/2} give D(E∗ 1/2) =D(E1/2)∗and D(E1/2) =2{D(E1/2) E1/2E1/2} +{E1/2D(E1/2) E1/2}, therefore D(E1/2)=−a0 0a J.M. Isidro / Journal of Algebra 635 (2023) 411–435 429 for some a ∈Rwith a =a∗. From {E1/2E11 E1/2} =E22 or {E1/2E22 E1/2} =E11 and (3)one gets a =2xand we finally get D(E11)=0x x0D(E1/2)=−2x0 02xD(E22)=0−x −x0(35) for some x ∈Rwith x∗=x. Let us define T∈M2by T:= 0x −x0, hence T∗= Tt=−T. Remark that T/∈S2however S2is Ad (T)-invariant since for A ∈S2we have At=A, hence T,At=TA−ATt=AtTt−TtAt=−AT +TA=T,A. Thus Ad (T)(S2) ⊂S2, and one shows, as in Example 2.3, that Ad (T) ∈Der (S2)and satisfies Ad (T)1=0, hence Δ := D+Ad (T)is also a quadratic Jordan derivation of S2. It is easy to check that Ad (T)A∗=Ad (T)(A∗)for A ∈S2, hence Δ ∈Der (S2)and enjoys the properties (9), (10), (11). Using the expressions of D(Eii)in (35), a simple matrix calculation gives Δ(Eii) =0for i =1, 2and therefore ΔQ(Eii)(A):=Δ{EiiAEii} ={Δ(Eii)AEii}+{EiiΔ(A)Eii}+{EiiAΔ(Eii)} ={EiiΔ(A)Eii}=Q(Eii)Δ(A) for all A ∈S2. Thus Δcommuteswith Q(Eii)for i =1, 2which completes the proof.  We would like to remark that also Δ(E1/2) =0holds. However, as noticed above, we have Q(E1/2)A =2{E11A∗E22}and as Δis a quadratic (but not a triple product) Jordan derivation we can not conclude Δ ◦Q(E1/2) =Q(E1/2) ◦Δ. This is the reason why from now on we assume that the given derivation δis additive. Next we shall find an explicit expression for Δ. 3.3 Lemma. Let δ∈Der (S2)be additive, and let Δ := δ−Jδ(1)+Ad (T)be the derivation of S2constructed in Lemma 3.2. Then there is an additive *-ring derivation φof Rsuch that φ(1) =0and Δ(A) =Aφholds for all A ∈S2. Proof. The elements A ∈S2and Δ(A)have the form A=a11 a1/2 a1/2a22 Δ(A)=b11 b1/2 b1/2b22  for some aii, a1/2, bii, b1/2∈R. From Q(Eii) ◦Δ =Δ ◦Q(Eii)and (33)we get biiEii =Q(Eii)Δ(A)=ΔQ(Eii)A=Δ(aiiEii) (36) 430 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 whence the (ii)-entry in Δ(A) depends only on the (ii)-entry of A. Clearly Jδ(1)and Ad (T)are additive and as we have assumed δto be additive, so is Δ. By Lemma 3.2, Δcommuteswith Q(Eii)and being additive also commutes with the sum Q(E1/2) = Id S2−Q(E11 +Q(E22). The above argument in (36)[now with E1/2instead of Eii] shows that the 1/2-entry in Δ(A) depends only on the 1/2- entry in A. Hence Δ(A)=φ1(a11)φ1/2(a1/2) φ1/2(a1/2)φ2(a22)(A∈S2) for some functions φi, φ1/2:R →R. Furthermore, from Δ(Eii) =0 =Δ(E1/2)for i =1, 2we conclude that φi(0) =0and φi(1) =0for i =1, 1/2, 2. Remark that the arguments made in the proof of Lemma 2.10 remain valid in the new situation since the particular elements involved in them (say J:= ΣEij now replaced with J:= E11 +E1/2+E22) actually belong to S2. This completes the proof.  3.4 Corollary. Let Rbe a unital *-ring that contains a complex unit and in which every element can be halved. Then each additive quadratic derivation δ∈Der (S2)is of the form δ(A):=Jδ(1)A+Ad (T)A+Aφ(A∈S2) where T∗=Tt=−T∈M2and φ ∈der(R)is additive with φ(1) =0. The representation of δis unique up to an element of Ad (T0)where T0∈M2commutes with all matrices in S2, Quadratic Jordan derivations in Sn: The induction step Our next task is the proof of Theorem 3.1 for which we need some preliminary material. For any matrix Am∈Smwe define Am⊕0∈Sm+1 to be the matrix Am⊕0:= Am0 0 0 . Clearly the map J:Sm→Sm+1 given by J(Am) := Am⊕0is a Jordan triple isomorphism from Sminto Sm+1. We define Pm+1 := Im⊕0. The map Pm+1 :Sm+1 →Sm+1 given by Pm+1(A) := Pm+1Ais a projection from Sm+1 into itself that satisfies Pm+1J(A)=A⊕0(A∈Sm). We identify each element A ∈Smwith its image J(A) ∈Sm+1. As in the proof of Lemma 2.13 one shows the following J.M. Isidro / Journal of Algebra 635 (2023) 411–435 431 3.5 Lemma. If Ψ ∈Der (Sm+1)satisfies Ψ(Pm+1) =0then for each A ∈Sm+1 there is a Bm∈Smuniquely determined by Asuch that Ψ(Pm+1APm+1)=Bm⊕0.(37) The map  Ψ: Sm→Smthat takes each Am∈Sminto the unique Bm∈Smfor which (16) holds satisfies  Ψ∈Der (Sm),  Ψ(Im) =0and ΨAm⊕0= Ψ(Am) ⊕0. If Ψis additive so is  Ψ. After these preliminaries we proceed to the induction proof of Theorem 3.1. Assume that (32)holds or some m ≥2; we have to show that then (32)also holds for m +1. Thus let Φ ∈Der (Sm+1)be given and assume that Φis additive. We have to prove the existence of an operator T∈Mm+1 that satisfies T∗=Tt=−Tand an additive * derivation φof the ring Rsuch that Φ(A)=JΦ(1)A+Ad (T)A+Aφ(A∈Sm+1).(38) Just as in the induction step in the proof of Theorem 2.1, we replace the given derivation Φ ∈Der (Sn) successively with Ψ := Φ −JΦ(1), then with Θ := Ψ −Ad (U)and after that with Δ := Θ −Ad (V)and remark that Φ, Ψ, Θand Δbelong to Der (Sm)and are additive. The proof goes on as follows: Step 3Claim For each fixed j∈1, 2, ··· , mthere is a matrix Dj∈Sm+1 such that D2 j=Im+1 D∗ j=DjPm+1Δ(Dj)Pm+1 =0. Proof. For each i ∈{1, ··· , m}we define Di:= Im+1 −Eii +(Ei,(m+1) +E(m+1),i) − E(m+1),(m+1), in details Di:= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1·0 1· · ··0··1 ·1· ·1 0 0·1·0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. Every matrix A ∈Sm+1 can be represented as a symmetric 2 ×2matrix in the form A=A11 A12 A21 A22  where A11 ∈Sm, A21 =At 12 and A22 is an element in R. By assumption there is some matrix xi:= (x1i, ·, xii, ·xim) ∈M1×m(R)such that Δ(Di) =0mxt i xizor in detail 432 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 Δ(Di)=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ·xi1 · · ··0m·· xii · · ·xim xi1·xii ·xim zi ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. From Δ(Di)2=Im+1 together with Δ(Im+1) =0and Δ(Di)2=DiΔ(Di) +Δ(Di)Di we get xi= (0) and z=0. In particular Δ(Di) =0for all i ∈{1, 2, ··· , m}and so by (22)we have Pm⊕0Δ(A)Pm⊕0)=Δ Pm⊕0APm⊕0 = ΔA11 ⊕0=(A11)φ⊕0.(39) Since by assumption is m ≥2, there is at least one j∈{1, ··· , m}such that j=i. If j is in those conditions then DjEijDj=Ei,(m+1) DjEjiDj=0 and DjEijDj=E(m+1),i DjEijDj=0 hold, hence by setting Sij := Eij +Eji we get DjSimDj=DjEij +EjiDj=Si,(m+1). Therefore for any a ∈R, from Δ(Di) =Δ(Dj) =0and (3)we obtain ΔaSi,(m+1)=Δ aDjSimDj=DjΔ(aSij)Dj(40) Since i, j≤m, by the induction hypothesis we have Δ(aSij) = Δ(aSij) =φ(a)Sij, which replaced above gives DjΔ(aSij)Dj=Dj Δ(aSij)Dj=φ(a)DjSijDj=φ(a)S(m+1),i .(41) On the other hand, from S(m+1),(m+1) =D1S11D1we get Δ(aS(m+1),(m+1))=Δ D1aS11 D1=D1φ(a)S11D1=φ(a)S(m+1),(m+1).(42) The relations (40), (41)and (42)imply Δ(aSij) =φ(a)Sij for i, j∈{1, ··· , m; m +1}, that is Δ(A) =φ(aij)=Aφfor all A ∈Sm+1. 4. Quadratic derivations on some classical Cartan factors Now we apply the results in the previous sections to the case in which the basic *-ring Ris C, the * operation being the complex conjugation. Thus Mnand Snare now the classical Cartan factors Inand IIn, respectively. The group Der (C)of quadratic Jordan derivations of Cwas fully described in [5] Lemma 2.3: J.M. Isidro / Journal of Algebra 635 (2023) 411–435 433 4.1 Lemma. Any additive δ∈Der (C)is of the form δ(a) =Jδ(1)(a), a ∈C. In particular, δis a continuous complex linear map on Cand if δ(1) =0then δis the null map. Hence by Theorems 2.1 and 3.1, each quadratic Jordan derivation δ∈Der (Mn), respectively δ∈Der (Sn)with δadditive, is of the form δ=Jδ(1)+Ad (T) for a suitable operator T∈Mn. Since Jordan multiplication Jδ(1)by a fixed element and Lie multiplication Ad (T)by a fixed element Tare automatically continuous complex linear maps, so is each quadratic Jordan derivation δof the mentioned Cartan factors. We conjecture that a similar result holds for Cartan factors of type III2n, though this will require a specific discussion. Quadratic Jordan derivations of Cartan factors of type IVnare studied below and the exceptional Cartan factors we dealt with in [4]. Derivations on Type IVnCartan factors Let Cn, (n ≥2), be endowed with its usual Hilbert space norm, and let A := L(Cn) denote the space of all complex linear maps A:Cn→Cnendowed with the natural operations and the operator norm. A Cartan factor of type IVn, also called a spin factor, is a complex vector subspace X ⊂L(Cn)with these two properties: (1) Xis selfadjoint i.e. if A ∈Xthen A∗∈X, and (2) if A ∈Xthen A2=λA1for some λA∈C. It is known (see [3]Proposition 16.6.16) that X, endowed with the usual triple product {ABC} := 1 2(AB∗C+CB∗A)and the operator norm, is a JBW∗-triple of rank 2. Therefore any maximal set of minimal pairwise orthogonal tripotents in Xconsists of two elements (p1, p2). Fix once and for all one such a pair (p1, p2), sometimes referred to as a frame in X. Then 1X:= p1+p2is a maximal tripotent in Xand Xbecomes a complex unital JB∗-algebra with unit 1X, the operator norm and the product “◦” and adjoint operation “#” given by A◦B:= {A1XB}A#:= {1XA1X}(A, B ∈X). In fact Xis special, that is, isomorphic to a Jordan subalgebra of AJ(the Jordan algebra associated with A), and any element A ∈Xcan be represented as a symmetric 2 ×2 matrix with complex entries A=a11 a12 a21 a22 . Let M2:= M2(C)and fix a pair F1, F2of orthogonal tripotents in M2with F1+F1=1.1 Let Xdenote the Jordan JB∗-subalgebra of M2generated by the pair (F1, F2); thus 1Thus the Fihave the same properties (but do not necessarily coincide) with the matrices Eiinvolved in the proof of Lemma 3.4. 434 J.M. Isidro / Journal of Algebra 635 (2023) 411–435 X ⊆M2though Xmay fail to be the whole of M2. By the Sirshov-Cohen theorem, Xis always a special Jordan algebra. Let δ:X →Xbe an additive quadratic Jordan derivation on X, and observe that the calculations and arguments that were made in the proof of Corollary 3.4 remain valid if we replace S2with Xand the tripotents Ei∈S2 with the tripotents Fi∈X, i =1, 2. Observe also that any type IVnCartan factor Xadmits a representation as the JB∗-subalgebra of M2generated by a suitable pair (F1, F2)of minimal orthogonal tripotents in Xwith F1+F2=1. Therefore we have the following version of Corollary 3.4 for type IVnCartan factors. 4.2 Corollary. If Xis a Cartan factor of type IVnthen each additive quadratic derivation Der (X)is of the form δ(A)=Jδ(1)A+Ad (T)A(A∈X),(43) where T∗=Tt=−T∈M2. The above representation is unique up to a term of the form Ad (T0)for some matrix T0∈M2that commutes with all elements in X. Data availability No data was used for the research described in the article. References [1] Lawrence A. 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