This paper is devoted to the study of Jordan isomorphisms on nest subalgebras of factor von Neumann algebras. It is shown that every Jordan isomorphism ϕ between the two nest subalgebras algMβ and algMγ is either an isomorphism or an anti-isomorphism.
1. Introduction
Let A and B be two associative algebras. A Jordan isomorphism ϕ from A onto B is a bijective linear map such that ϕ(T2)=ϕ(T)2 for every T∈A. Obviously, isomorphisms and anti-isomorphisms are basic examples of Jordan isomorphisms. Jordan isomorphisms have been studied by many authors for various rings and algebras (see [1–5]). The standard problem is to determine whether a Jordan isomorphism is either an isomorphism or an anti-isomorphism. Using linear algebraic techniques, Molnár and Šemrl [6] proved that automorphisms and antiautomorphisms are the only linear Jordan automorphisms of Tn(F), n⩾2, where F is a field with at least three elements. Later, Beidar et al. [7] generalized this result and proved that every linear Jordan isomorphisms of Tn(C), n⩾2, onto an arbitrary algebra over C, is either an isomorphism or an anti-isomorphism, where C is a 2-torsionfree commutative ring which is connected, that is, a ring in which the only idempotents are 0 and 1. Recently, Zhang [8] proved that every Jordan isomorphism ϕ between two nest algebras algβ and algγ is either an isomorphism or an anti-isomorphism. The same result was concluded in [9] for Jordan isomorphisms on nest algebras. The motivation for this paper is the work by Zhang. The aim of the present paper is to characterize Jordan isomorphisms of nest subalgebras.
In fact, the characterization of Jordan isomorphisms is closely related to isometric problems (see [10–12]). However, Jordan algebras or structures are related to vertex (operator) algebras or superalgebras and to representations of Kac-Moody and Virasoro algebras [13]. Note that the vertex operators of string theory also give reps of Lie algebras and Kac-Moody and Virasoro algebras (both infinite dimensional Lie algebras) as well as reps of the Fischer-Griess Monster group algebra. Jordan algebras or structures are also related to QFT, CFT, SCAs, and WZW models [14]. Another paper [15] about M-theory with J3(O) and F4 makes use of the algebraic Kostant-Dirac operator, and a variant of this operator also appears on page 12 of a paper [16] about a gerbe based approach to supersymmetric WZW models (gerbes are useful in string theory; e.g., gerbes help in illuminating the geometry of mirror symmetry of CY threefolds, help in giving a noncommutative description of D-branes in the presence of topologically nontrivial background fields, and provide a geometric way to unify properties of p-form fields with gauge symmetries, etc.). There is more about how Jordan structures are involved with physics [17]. This paper studies the structure of Jordan isomorphisms of nest subalgebras. Our main result is that every Jordan isomorphism of nest subalgebras is either an isomorphism or an anti-isomorphism.
Now, let us introduce some notions and concepts that will be used later. Let M be a von Neumann algebra acting on a separable Hilbert space H. A nest β in M is totally a family of (self-adjoint) projections in M which is closed in the strong operator topology and included 0 and I. The nest subalgebra of M associated with a nest β, denoted by algMβ, is the set algMβ={T∈M:PTP=TPforallP∈β}. The diagonal DM(β) of a nest subalgebra algMβ is the von Neumann subalgebra (algMβ)∩(algMβ)*. Let RM(β) denote the norm closed algebra generated by {PT(I-P):T∈M,P∈β}. It is clear that RM(β) is a norm closed ideal of the nest subalgebra algMβ. If M is a factor von Neumann algebra, it follows from [18] that DM(β)+RM(β) is weakly dense in algMβ. When M=B(H), algMβ is called a nest algebra and is denoted by algβ. As a notational convenience, if P is an idempotent, we let P⊥ denote I-P throughout this paper.
We refer the readers to [19] for background information about von Neumann algebras and to [18] for the theory of nest algebras and nest subalgebras.
2. Main Result
The following theorem is our main result.
Theorem 1.
Let algMβ, algMγ be two nontrivial subalgebras in factor von Neumann algebra M, and let ϕ:algMβ→algMγ be a Jordan isomorphism. Then, ϕ is either an isomorphism or an anti-isomorphism.
The proof is purely algebraic and will be organized in a series of lemmas. As a notational convenience, throughout this paper, we assume that β and γ are nontrivial nests in a factor von Neumann algebra M and that ϕ:algMβ→algMγ is a Jordan isomorphism.
Lemma 2.
Let β be a nest in factor von Neumann algebra M, and then, for X,Y∈M, X(algMβ)Y=0 if and only if there exists a projection P∈β such that X=XP⊥, Y=PY.
Proof.
It is clear that if there exists a projection P∈β such that X=XP⊥, Y=PY, then X(algMβ)Y=0.
Clearly, let K be the closure of the space (algMβ)YH, and let P be the projection onto K. Then, P∈M and P⊥TP=0 for all T∈algMβ, and so P∈β. It is clear that XP=0, P⊥Y=0. Thus, X=XP⊥, Y=PY. The proof is complete.
Lemma 3.
ϕ(I)=I, and, for any A,B,T∈algMβ, the following are equivalent:
ϕ(A2)=ϕ(A)2;
ϕ(AB+BA)=ϕ(A)ϕ(B)+ϕ(B)ϕ(A);
ϕ(ABA)=ϕ(A)ϕ(B)ϕ(A);
ϕ(ABT+TBA)=ϕ(A)ϕ(B)ϕ(T)+ϕ(T)ϕ(B)ϕ(A).
Proof.
As (a)⇒(b), let X=A+B, and then, by (a), we have (1)φX2=ϕA2+ϕAB+BA+ϕB2=ϕA+ϕB2.This shows that (b) is established.
As (b)⇒(c), by (b), (2)2ϕABA=ϕ[(AB+BA)A+A(AB+BA)]-ϕ(A2B+BA2)=[ϕ(A)ϕ(B)+ϕ(B)ϕ(A)]ϕ(A)=+ϕAϕAϕB+ϕBϕA=-ϕA2ϕ(B)-ϕ(B)ϕA2=2ϕ(A)ϕ(B)ϕ(A).
As (c)⇒(d),(3)ϕABT+TBA=ϕA+TBA+T-ϕABA-ϕTBT=ϕA+ϕTϕBϕA+ϕT=-ϕ(A)ϕ(B)ϕ(A)-ϕTϕBϕT=ϕ(A)ϕ(B)ϕ(T)+ϕTϕBϕA.
Taking B=I in (b), we get (4)2ϕ(A)=ϕ(A)ϕ(I)+ϕ(I)ϕ(A).Multiplying the above equation on both sides by ϕ(I) and noticing that ϕ(I)2=ϕ(I), we have (5)ϕIϕA=ϕIϕAϕI,ϕ(A)ϕ(I)=ϕ(I)ϕ(A)ϕ(I).Thus, for any A∈algMβ, ϕ(A)ϕ(I)=ϕ(I)ϕ(A). It follows from M∩(algMβ)′=CI that ϕ(I)=I. Hence, (a)–(d) are equivalent.
Lemma 4.
If P∈β/{0,I}, then, for all T,S∈algMβ, one has(6)ϕ(P)ϕ(T)ϕ(P⊥)ϕ(S)ϕ(P)=0;ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)=0;ϕ(P)ϕ(S)ϕ(P⊥)ϕ(T)ϕ(P)=0;ϕ(P⊥)ϕ(T)ϕ(P)ϕ(S)ϕ(P⊥)=0.
Proof.
By Lemma 3(b), for all T∈algMβ,(7)ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥)+ϕ(P⊥)ϕ(T)ϕ(P).Let S∈algMβ, and then (8)ϕPϕTϕP⊥ϕSϕP+ϕ(P⊥)ϕ(T)ϕ(P)ϕ(S)ϕ(P⊥)+ϕPϕSϕP⊥ϕTϕP+ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)=ϕ(P)ϕ(T)ϕ(P⊥)+ϕ(P⊥)ϕ(T)ϕ(P)·ϕ(P)ϕ(S)ϕ(P⊥)+ϕ(P⊥)ϕ(S)ϕ(P)+ϕPϕSϕP⊥+ϕP⊥ϕSϕP·ϕPϕTϕP⊥+ϕP⊥ϕTϕP=ϕ(PTP⊥)ϕ(PSP⊥)+ϕ(PSP⊥)ϕ(PTP⊥)=0.This implies that (9)ϕPϕTϕP⊥ϕSϕP+ϕPϕSϕP⊥ϕTϕP=0,ϕP⊥ϕTϕPϕSϕP⊥+ϕP⊥ϕSϕPϕTϕP⊥=0.By (9),(10)ϕ-1ϕ(P)ϕ(T)ϕ(P⊥)ϕ(S)ϕ(P)+ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)=ϕ-1ϕPϕTϕP⊥ϕ-1ϕP⊥ϕSϕP+ϕ-1ϕP⊥ϕSϕPϕ-1ϕPϕTϕP⊥=PTP⊥-ϕ-1ϕP⊥ϕTϕP·PSP⊥-ϕ-1ϕPϕSϕP⊥+PSP⊥-ϕ-1ϕPϕSϕP⊥·PTP⊥-ϕ-1ϕP⊥ϕTϕP=ϕ-1ϕ(P⊥)ϕ(T)ϕ(P)ϕ(S)ϕ(P⊥)+ϕ(P)ϕ(S)ϕ(P⊥)ϕ(T)ϕ(P)+PAP⊥=-ϕ-1ϕ(P)ϕ(T)ϕ(P⊥)ϕ(S)ϕ(P)+ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)+PAP⊥,where (11)A=-PTP⊥ϕPϕSϕP⊥ϕ-1ϕP⊥ϕTϕP=-+ϕ-1ϕP⊥ϕTϕPPSP⊥=-+ϕ-1ϕPϕSϕP⊥PTP⊥=-+PSP⊥ϕ-1ϕP⊥ϕTϕP.Clearly, A∈algMβ. Thus, by the above equation, we have (12)ϕPϕTϕP⊥ϕSϕP+ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)=12ϕPAP⊥=12[ϕ(P)ϕ(A)ϕ(P⊥)+ϕ(P⊥)ϕ(A)ϕ(P)].This shows that, for any T,S∈algMβ,(13)ϕPϕTϕP⊥ϕSϕP=ϕ(P⊥)ϕ(S)ϕ(P)ϕ(T)ϕ(P⊥)=0.
Lemma 5.
For any T∈algMβ and any projection P∈β, either ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥) or ϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P).
Proof.
If P=0 or P=I, the result is clear. Suppose that P∈β/{0,I}. Let (14)A=ϕPϕTϕP⊥,B=ϕ(P⊥)ϕ(T)ϕ(P).Then, by Lemma 4, for any X∈algMγ,(15)ϕ(P⊥)XA=ϕ(P)XB=0.By Lemma 2 and (15), there exists a projection P1∈γ such that (16)ϕ(P⊥)=ϕ(P⊥)P1⊥;(17)A=P1A.There exists a projection P2∈γ such that (18)ϕ(P)=ϕ(P)P2⊥;(19)B=P2B.By (16) and (18), we have (20)ϕ(P⊥)P1⊥+ϕ(P)P2⊥=ϕ(I)=I.Multiplying the above equation on both sides from the right side by P1P2, we get P1P2=0. If A≠0, then by (17) P1≠0. So, P2=0. Hence, by (19), B=0. Similarly, if B≠0, then A=0. This implies that A=0 or B=0. From the fact that ϕ(PTP⊥)=A+B, for all T∈algMβ, one of the following is set up: (21)ϕPTP⊥=ϕPϕTϕP⊥,ϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P).Since M is factor, then there exists a partial isometry operator V∈M such that V=PVP⊥, thus, V∈algMβ. Therefore, either ϕ(V)=ϕ(P)ϕ(V)ϕ(P⊥) or ϕ(V)=ϕ(P⊥)ϕ(V)ϕ(P). Suppose ϕ(V)=ϕ(P)ϕ(V)ϕ(P⊥), if there exists S∈algMβ such that ϕ(PSP⊥)≠ϕ(P)ϕ(S)ϕ(P⊥), then ϕ(PSP⊥)=ϕ(P⊥)ϕ(S)ϕ(P). On the other hand, one of the following is set up: (22)ϕV+PSP⊥=ϕPϕV+SϕP⊥,ϕ(V+PSP⊥)=ϕ(P⊥)ϕ(V+S)ϕ(P).And ϕ(V)=ϕ(P)ϕ(V)ϕ(P⊥), ϕ(PSP⊥)≠ϕ(P)ϕ(S)ϕ(P⊥); hence,(23)ϕ(V+PSP⊥)=ϕ(P⊥)ϕ(V+S)ϕ(P).So, ϕ(V)=ϕ(P⊥)ϕ(V)ϕ(P)=ϕ(P)ϕ(V)ϕ(P⊥). This shows that ϕ(V)=0; thus, V=0. A contradiction. In conclusion, for any T∈algMβ, we have ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥). Similarly, suppose that ϕ(V)=ϕ(P⊥)ϕ(V)ϕ(P), and then, for any T∈algMβ, we have ϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P). Consequently, for any T∈algMβ and any projection P∈β, either ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥) or ϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P).
Proof of Theorem 1.
By Lemma 5, if, for any T∈algMβ,(24)ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥),let A∈algMβ, and then (25)ϕPAPTP⊥=ϕ(P)ϕ(PAP)ϕ(PTP⊥)ϕ(P⊥)+ϕ(P)ϕ(PTP⊥)ϕ(PAP)ϕ(P⊥).Thus, for all T,A∈algMβ, we have (26)ϕ(PAPTP⊥)=ϕ(PAP)ϕ(PTP⊥).Clearly, (27)ϕ(PAP⊥PTP⊥)=ϕ(PAP⊥)ϕ(PTP⊥)=0,ϕ(P⊥AP⊥PTP⊥)=ϕ(P⊥AP⊥)ϕ(PTP⊥)=0.By (26)-(27), for all A,T∈algMβ, we have (28)ϕ(APTP⊥)=ϕ(A)ϕ(PTP⊥).Similar to the proof of (28), for any A,T∈algMβ,(29)ϕ(PTP⊥A)=ϕ(PTP⊥)ϕ(A).Let A,B∈algMβ, and by (28), for any T∈algMβ,(30)ϕABPTP⊥=ϕABϕPTP⊥=ϕAϕBPTP⊥=ϕ(A)ϕ(B)ϕ(PTP⊥).Thus, for all A,B∈algMβ, we have (31)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)algMγϕ(P⊥)=0.Similarly, by (29), (32)ϕ(P)algMγϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)]=0.By the above two equations and Lemma 2, there exists a projection Q1∈γ such that (33)ϕAB-ϕAϕBϕP=[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)Q1⊥;(34)ϕ(P⊥)=Q1ϕ(P⊥).And there exists a projection Q2∈γ such that (35)ϕ(P)=ϕ(P)Q2⊥;(36)ϕP⊥ϕAB-ϕAϕB=Q2ϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)].Since ϕ is a Jordan isomorphism and ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥), then, for any T∈algMβ, we have (37)ϕ(P⊥)ϕ(T)ϕ(P)=0.Especially, ϕ(P⊥)Q1⊥ϕ(P)=0; that is, ϕ(P⊥)Q1⊥ϕ(P⊥)=ϕ(P⊥)Q1⊥. Thus, by formula (34), we have (38)ϕ(P⊥)Q1⊥=Q1⊥ϕ(P⊥)=0.This implies that Q1⊥[ϕ(P⊥)]=0, where [ϕ(P⊥)] is an orthogonal projection onto ϕ(P⊥)H. Since P is a nontrivial projection, then [ϕ(P⊥)]≠0, thus, Q1⊥=0. Therefore, by (33), (39)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)=0.By (35) and similar discussion, we get (40)ϕ(P)Q2=Q2ϕ(P)=0.This shows that [ϕ(P)]Q2=0; thus, Q2=0. So, by (36), (41)ϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)]=0.In addition, by Lemma 3(c), we have (42)ϕPϕABϕP⊥=ϕPAPBP⊥+ϕPAP⊥BP⊥=ϕ(PAP)ϕ(PBP⊥)+ϕ(PAP⊥)ϕ(P⊥BP⊥)=ϕPϕAϕPϕBϕP⊥+ϕ(P)ϕ(A)ϕ(P⊥)ϕ(B)ϕ(P⊥).Thus, for any A,B∈algMβ, we have (43)ϕ(P)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P⊥)=0.By (39) and (41), for all A,B∈algMβ, we have ϕ(AB)=ϕ(A)ϕ(B). Hence, ϕ is an isomorphism.
If for any T∈algMβ, we have ϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P), then, for any X∈algMβ⊥, we define ψ(X)=ϕ(JX*J), where J is a conjugate linear involution operator defined in Lemma 2.3 of [11]. It is not difficult to verify that ψ:algMβ⊥→algMγ is a Jordan isomorphism and, for any projection Q∈β⊥/{0,I}, we have (44)ψ(QXQ⊥)=ψ(Q)ψ(X)ψ(Q⊥).Thus, from the above discussion, ψ is an isomorphism. Consequently, ϕ is an anti-isomorphism.
By Lemma 5, for all T∈algMβ, one of the following holds: (45)ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥)orϕ(PTP⊥)=ϕ(P⊥)ϕ(T)ϕ(P).If ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥), then Lemma 3(c) and the fact that BP=PBP imply that (46)ϕABϕPTP⊥=ϕABPTP⊥=ϕAϕBPTP⊥=ϕ(A)ϕ(B)ϕ(PTP⊥)for all A,B∈algMβ. Hence, for all A,B∈algMβ, we have (47)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)algMγϕ(P⊥)=0.Similarly, it follows from Lemma 3(c) and P⊥A=P⊥AP⊥ that, for all A,B∈algMβ, we have (48)ϕ(P)algMγϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)]=0.By (47), (48), and Lemma 2, there exists a projection Q1,Q2∈γ such that (49)ϕAB-ϕAϕBϕP=[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)Q1⊥;(50)ϕP⊥=Q1ϕP⊥;(51)ϕ(P)=ϕ(P)Q2⊥;(52)ϕP⊥ϕAB-ϕAϕB=Q2ϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)].Since ϕ(PTP⊥)=ϕ(P)ϕ(T)ϕ(P⊥), we have ϕ(P⊥)ϕ(T)ϕ(P)=0 for all T∈algMβ. Let ϕP∈M, and [ϕ(P)] denote the orthogonal projection from H to ϕ(P)H. In particular, ϕ(P⊥)Q1⊥ϕ(P)=0; that is, ϕ(P⊥)Q1⊥ϕ(P⊥)=ϕ(P⊥)Q1⊥. Thus, by formula (50), (53)ϕ(P⊥)Q1⊥=Q1⊥ϕ(P⊥)=0.This shows that Q1⊥[ϕ(P⊥)]=0. Since P is a nontrivial projection, then [ϕ(P⊥)]≠0, so Q1⊥=0. By (49), (54)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P)=0.In similar discussion, we have ϕ(P)Q2=Q2ϕ(P)=0. This shows that [ϕ(P)]Q2=0, so Q2=0. Thus, by (52), (55)ϕ(P⊥)[ϕ(AB)-ϕ(A)ϕ(B)]=0.Because P⊥BP⊥PAP⊥=0, by Lemma 3, we have (56)ϕPAP⊥BP⊥=ϕP⊥BP⊥∘PAP⊥=ϕP⊥BP⊥∘ϕPAP⊥=ϕ(P⊥BP⊥)ϕ(PAP⊥)for all A,B∈algMβ. Since PBP⊥PAP=0, similarly, (57)ϕ(PAPBP⊥)=ϕ(PAP)ϕ(PBP⊥).By (56), (57), and Lemma 4, (58)ϕPϕABϕP⊥=ϕPAPBP⊥+ϕPAP⊥BP⊥=ϕ(PAP)ϕ(PBP⊥)+ϕ(PAP⊥)ϕ(P⊥BP⊥)=ϕPϕAϕPϕBϕP⊥+ϕ(P)ϕ(A)ϕ(P⊥)ϕ(B)ϕ(P⊥)=ϕ(P)ϕ(A)ϕ(B)ϕ(P⊥).Thus, for all A,B∈algMβ, we have (59)ϕ(P)[ϕ(AB)-ϕ(A)ϕ(B)]ϕ(P⊥)=0.By (54), (55), and (59), we have ϕ(AB)=ϕ(A)ϕ(B) for any A,B∈algMβ. Similarly, ϕ(AB)=ϕ(B)ϕ(A).
Consequently, ϕ is either an isomorphism or anti-isomorphism.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author wishes to thank the referees for their time and comments. He thanks Dr. Yongjian Xie from the College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China. This paper is supported by the Fundamental Research Funds for the Central Universities (Grant no. GK201503017).
HersteinI. N.Jordan homomorphisms195681331341MR007675110.1090/S0002-9947-1956-0076751-6BaxterW. E.MartindaleW. S.Jordan homomorphisms of semiprime rings19795624574712-s2.0-000133971210.1016/0021-8693(79)90349-1MR528587BrešarM.Jordan mappings of semiprime rings1991442233238MR112636110.1017/s000497270002966xBrešarM.Jordan mappings of semiprime rings1989127121822810.1016/0021-8693(89)90285-8MR10294142-s2.0-0001470552WongT.-L.Jordan isomorphisms of triangular rings200513311338133882-s2.0-27844486429MR216116310.1090/s0002-9939-05-07989-xMolnárL.ŠemrlP.Some linear preserver problems on upper triangular matrices1998452-3189206MR16716192-s2.0-000054101410.1080/03081089808818586BeidarK. I.BrešarM.ChebotarM. A.Jordan isomorphisms of triangular matrix algebras over a connected commutative ring20003121–319720110.1016/s0024-3795(00)00087-2MR1759333ZhangJ. H.Jordan isomorphisms of nest algebras2002454819824MR1925327LuF.Jordan isomorphisms of nest algebras2003131114715410.1090/s0002-9939-02-06587-5MR1929034ChebotarM. A.KeW.-F.LeeP.-H.WongN.-C.Mappings preserving zero products20031551779410.4064/sm155-1-6MR19611622-s2.0-0037698821HouJ.JiaoM.Additive maps preserving Jordan zero-products on nest algebras2008429119020810.1016/j.laa.2008.02.021MR24191492-s2.0-42749095668ZhangJ.-H.YangA.-L.PanF.-F.Linear maps preserving zero products on nest subalgebras of von Neumann algebras20064122-334836110.1016/j.laa.2005.07.001MR21829662-s2.0-27944482950XuX.Simple conformal algebras generated by Jordan algebrashttp://arxiv.org/abs/math/0008224GünaydinM.Extended superconformal symmetry, Freudenthal triple systems and gauged WZW models1995447Berlin, GermanySpringer5469Lecture Notes in Physics10.1007/3-540-59163-x_255MR1356233RamondP.Algebraic dreamshttp://arxiv.org/abs/hep-th/0112261MickelssonJ.Gerbes, (twisted) K-theory, and the supersymmetric WZW modelhttp://arxiv.org/abs/hep-th/0206139GauntlettJ. P.GibbonsG. W.HullC. M.TownsendP. K.BPS states of D = 4 N = 1 supersymmetry2001216243145910.1007/s002200000341GilfeatherF.LarsonD. R.Nest-subalgebras of von Neumann algebras198246217119910.1016/0001-8708(82)90022-6MR6799072-s2.0-0000038135DavidsonK. R.1988191Longman Scientific & TechnicalPitman Research Notes in MathematicsMR972978