TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 3, March 1998, Pages 1277–1284 S 0002-9947(98)02102-3

A CLASSIFICATION THEOREM FOR ALBERT ALGEBRAS

R. PARIMALA, R. SRIDHARAN, AND MANEESH L. THAKUR

Abstract. Let k be a field whose characteristic is different from 2 and 3 and let L/k be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra B over L with an involution σ of the second kind over k, the Jordan algebra J(B, σ, u, µ), obtained through Tits’ second construction is determined up to isomorphism by the class of (u, µ)inH1(k, SU(B, σ)), thus settling a question raised by Petersson and Racine. As a consequence, we derive a “Skolem Noether” type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of J(B, σ, u, µ), if (B, σ) is fixed.

Introduction Let k be a field with characteristic different from 2 and 3. Exceptional simple Jordan algebras over k are called Albert algebras. There are rational constructions due to Tits of all Albert algebras over k, referred to as the first and the second constructions. We begin by recalling the second construction briefly. Let L/k be a quadratic extension, bar denoting its nontrivial k-automorphism. Let B be a degree 3 central simple (associative) algebra over L with an involution σ of the second kind over k.Letube a unit of B with σ(u)=uand N(u)=µµ, µ L, where N denotes the reduced norm on B.Let (B,σ)= x Bσ(x)=x .Let∈ J(B,σ,u,µ)= (B,σ) B be the Jordan algebraH with the{ multiplication∈ | given} by H ⊕ 1 (b ,b)(b0 ,b0)=(b .b0 + buσ^(b )+b^uσ(b), b b0 + b b + µ(σ(b) σ(b0))u− ), 0 0 0 0 0 0 0 0 × where x.y = 1 (xy + yx), x = 1 (t(x) x)and 2 2 − e e 1 1 1 x y=x.y t(x)y t(y)x + (t(x)t(y) t(x y)), × e− 2 − 2 2 − · t denoting the reduced trace of B. The cubic norm of an element (a0,a)of J(B,σ,u,µ)isgivenby ( ) n(a,a)=N(a )+µN(a)+µN(σ(a)) t(a auσ(a)). ∗1 0 0 − 0 For any unit w B, we have an isomorphism ∈ J(B,σ,u,µ) J ( B,σ,wuσ(w),N(w)µ) −→ given by (a ,a) (a ,aw). The following converse problem is posed in [P-R]. 0 7→ 0 Question. If J(B,σ,u,µ) J(B,σ,u0,µ0), then does there exist a unit w B ' ∈ such that u0 = wuσ(w),µ0=N(w)µ?

Received by the editors June 12, 1996. 1991 Mathematics Subject Classification. Primary 17C40. Key words and phrases. Albert algebras, Tits’ constructions.

c 1998 American Mathematical Society

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In this paper we answer this question in the affirmative ( 2, Theorem 2.7). As a consequence, we prove a “Skolem Noether” type theorem§ on extensions of iso- morphisms on certain types of simple Jordan subalgebras of an Albert algebra ( 3, 3 § Theorem 3.1). There are cohomological invariants g3 H (k, Z/3) and f3,f5in ∈ H3(k, Z/2) and H5(k, Z/2) respectively, attached to an Albert algebra ([S]). Serre raised the question whether these invariants determine the isomorphism class of the Albert algebra. We indeed show that if J(B,σ,u,µ)andJ(B,σ,u0,µ0)havethe same invariants, then they are isomorphic ( 2, Theorem 2.8). § 1. Coordinatization of a certain Tits second construction

Let L/k be a quadratic extension. Let denote the involution on M3(L)given 1 t ∗ by X ∗ =Γ− X Γ, where Γ = γ ,γ ,γ ,γ kwith γ γ γ = 1 and bar denoting h 1 2 3i i∈ 1 2 3 the entrywise action of the automorphism bar of L.LetV GL (L) with V ∗ = V . ∈ 3 Suppose further that det V = µµ for some µ L∗. Then one has the second Tits’ ∈ construction J(M3(L), ,V,µ) with the underlying space (M3(L), ) M3(L). ∗ 1 t H ∗ ⊕ The matrix U = V Γ− is hermitian, i.e, U = U.Further,det U = det V = µµ.Lethdenote the hermitian form on L3 given by h(x, y)=xUyt. Then the discriminant of h denoted by disc h is trivial. Let ψ :( 3L3, 3h) (L, 1 )be ' h i the trivialization of disc h given by e1 e2 e3 µ, ei being the standard basis vectors of L3. We then have the Cayley∧ algebra∧ (cf.7→ [T]), VC = C(VL3,h,ψ)=L L3, with multiplication defined by ⊕

(a, v)(a0,v0)=(aa0 h(v, v0),av0+av+θ(v, v0)), − 0 where θ is defined by the identity

h(v00,θ(v, v0)) = ψ(v00 v v0), ∧ ∧ 3 for all v, v0,v00 L . Also, the norm n is given by n (a, v)=n (a)+h(v), ∈ C C L/k where h(v)=h(v, v). Then one has the reduced Albert algebra 3(C, Γ) which consists of all 3 3 matrices H × 1 α1 cγ1−γ3b 1 X = γ2−γ1cα2 a ,  1  bγ3−γ2aα3 where α k, a, b, c C, with the multiplication (X, Y) 1 (XY + YX). Here i ∈ ∈ 7→ 2 the bar denotes the involution on C = L L3 given by (α, v)=(α, v). Note ⊕ − that 3(C, Γ) contains 3(L, Γ) = (M3(L), ) as a Jordan subalgebra. The cubic normH of any element XHin (C, Γ)H as above,∗ is given by H3 ( 2) ∗ 1 1 1 n(X)=α α α γ− γ α n (a) γ− γ α n (b) γ− γ α n (c)+t ((ca)b), 1 2 3 − 3 2 1 C − 1 3 2 C − 2 1 3 C C where, for (α, v) C, t (α, v)=t (α). With this notation we have the following. ∈ C L/k Theorem 1.1. The map Φ:J(M3(L), ,V,µ) 3(C, Γ), induced by the natural inclusion (M (L), ) , (C, Γ) and∗ the map→HM (L) (C, Γ) given by H 3 ∗ →H3 3 →H3 1 1 a1 0 γ1− a3 γ1− a2 1 − 1 a2 γ2− a3 0 γ2− a1 ,   7→  1 1 −  a3 γ− a γ− a 0 − 3 2 3 1     is an isomorphism of Jordan algebras, a1, a2, a3 denoting the rows of a matrix in M3(L).

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Proof. Clearly Φ is an isomorphism of vector spaces. Hence by ([MC-1], p. 507), to check that Φ is an isomorphism of Jordan algebras, it suffices to check that it preserves the cubic norms and maps identity to identity. It is clear from the definition that Φ(1) = 1. Let 1 α1 cγ1−γ3b a1 1 (A0,A)= γ−2γ1cα2 a ,a2  1   bγ3−γ2aα3 a3    be any element of J(M3(L), ,V,µ)= 3(L, Γ) M3(L). Equating the norms of (A ,A)andΦ(A,A)(cf.( ∗), ( )), weH need to⊕ verify that 0 0 ∗1 ∗2 det A + µdet A + µdet A∗ t(A AV A∗)=α α α 0 − 0 1 2 3 1 1 1 1 1 1 γ3− γ2α1nC(a, γ2− a1) γ1− γ3α2nC (b, γ3− a2) γ2− γ1α3nC (c, γ1− a3) − − − 1 −1 − 1 − +t (((c, γ− a )(a, γ− a ))(b, γ− a )) C − 1 3 − 2 1 − 3 2 i.e., 1 det A + µdet A + µdet A∗ t(A AV A∗)=α α α γ− γ α n(a) γ α h(a ) 0 − 0 1 2 3− 3 2 1 − 1 1 1 1 1 γ1− γ3α2n(b) γ2α2h(a2) γ2− γ1α3n(c) γ3α3h(a3)+tC((ca γ3h(a3, a1), − − 1 −1 − 1 − γ− ca γ− a¯a¯ + γ θ(a , a ))(b, γ− a )). − 2 1 − 1 3 3 3 1 − 3 2 We have 1 1 1 det A = α α α γ− γ n(a) γ− γ α n(b) γ− γ α n(c)+t (cab). 0 1 2 3 − 3 2 − 1 3 2 − 2 1 3 L/k Further, tC (α, v)=tL/k(α), so that we are reduced to verifying the following equality: ( ) ∗3 µdet A + µdet A∗ t(A AV A∗)= α γ h(a ) α γ h(a ) α γ h(a ) − 0 − 1 1 1 − 2 2 2 − 3 3 3 γ t (h(a , a )b) γ t (h(ca , a )) − 3 L/k 3 1 − 1 L/k 1 2 γ t (h(¯aa¯ , a )) + t (h(θ(a , a ), a )). − 2 L/k 3 2 L/k 3 1 2 We first compute t(A0AV A∗). We have 1 t t AV A∗ = AV Γ− A Γ=AUA Γ=(aij )Γ,

where aij = h(aj , ai). This gives

t(A0AV A∗)=α1γ1h(a1)+γ1ch(a1, a2)+γ3bh(a1, a3)+γ1ch(a2, a1)

+α2γ2h(a2)+γ2ah(a2, a3)+γ3bh(a3, a1)+γ2ah(a3, a2)+α3γ3h(a3). Comparing the above with ( ), it only remains to verify that ∗3 µdet A + µdet A∗ = tL/k(h(θ(a3, a1), a2)). By definition of θ,wehave h(θ(a,a),a)=ψ(a a a )=µdet A = µdet A. 3 1 2 2∧ 3∧ 1 Hence

tL/k(h(θ(a3, a1), a2)) = µdet A + µdet A = µdet A + µdet A∗.

For a Jordan algebra J 3(C, Γ),Cis called the coordinate algebra of J.Its isomorphism class is uniquely'H determined by J (cf. [A-J]).

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Corollary 1.2. Let C be any coordinatization of J(M (L), ,V,µ). The norm form 3 ∗ nC of C is given by nC = trL/k( 1 h),wherehis the hermitian form given by 1 h i⊥ the matrix V Γ− . Proof. By the uniqueness of the coordinate algebra of a reduced Albert algebra (cf. [A-J]) and by Theorem 1.1 we get C C(L3,h,ψ). The corollary now follows (cf. [T], p. 5124). '

2. Classification of Albert algebras Let k be a field with characteristic different from 2 and 3. Let J be an Albert algebra over k. Then there exist a 3-fold Pfister form φ3 and a 5-fold Pfister form φ5 over k such that Q φ 2,2,2 φ , J ⊥ 3 'h i⊥ 5 QJ denoting the trace quadratic form of J. This property characterizes φ3 and φ5 up to isomorphism (cf. [S]). Let G be the group of automorphisms of the split Albert algebra over k. We have the mod 2 cohomological invariants 1 3 f3 : H (k, G) H (k, Z/2) → and 1 5 f5 : H (k, G) H (k, Z/2), → defined as the Arason invariants of φ and φ respectively. If J (C, Γ), then 3 5 'H3 f3(J) is the Arason invariant of nC and f5(J) is the Arason invariant of nC 1 1 ⊗ 1,γ1− γ2 1,γ2− γ3 . Following a suggestion of Serre, Rost ([R]) constructed a hmod 3 invarianti⊗h i 1 3 g3 : H (k, G) H (k, Z/3). → If J = J(B,σ,u,µ) is a second Tits’ construction Albert algebra corresponding to a 3 3 quadratic extension L of k,theng3(J) H (k, Z/3) maps to [B] [µ] H (L, Z/3); more precisely, ∈ ∪ ∈ g (J)= cor ([B] [µ]). 3 − L/k ∪ We begin by reviewing a result for Albert algebras belonging to Tits’ first con- struction and which motivated the question posed in the introduction. Let k be a field of characteristic different from 2 and 3. Let A be a degree 3 central simple (associative) algebra over k and let µ k∗.LetAi=Afor i =0,1,2. On the k vector space J(A, µ)=A A A , we∈ define a multiplication by − 0⊕ 1⊕ 2

(a0,a1,a2)(a0,a10 ,a20 )=(a0.a0 + a1a20 + a10 a2, a0a10 + a0a1 1 +µ− a a0 ,aa +a0a +µa a0 ), 2 × 2 2 0 2g0 g1 × 1e e where x.y, x, x y are defined as in the introduction. With this multiplication, × e J(A, µ) is an Albert algebra. Given µ, µ0 ke∗ with µ0 = N(w)µ for some unit ∈ w A, theree is an isomorphism of J(A, µ) with J(A, µ0) given by (a0,a1,a2) ∈ 1 1 7→ (wa0w− ,wa1,a2w− ). Using the Rost invariant for Albert algebras, Petersson and Racine ([P-R]) proved that the converse is true, i.e., if J(A, µ) J(A, µ0), then ' there exists a unit w A such that µ0 = N(w)µ. ∈ Let L/k be a quadratic extension and J(B,σ,ui,µi) be the Tits’ second con- struction with respect to units ui B,σ(ui)=ui and µi L∗ with N(ui)=µiµi. The map ∈ ∈ ( (B,σ) B) L B B B , H ⊕ ⊗k → 0 ⊕ 1 ⊕ 2

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given by (a ,a) λ (λa ,λa, λu σ(a)), 0 ⊗ 7→ 0 i where B ,B ,B are copies of B, gives an isomorphism J(B,σ,u ,µ ) L 1 2 3 i i ⊗k ' J(B,µi), i =1,2. Thus, in view of the above discussion for algebras of the first kind, we have

Proposition 2.1 (cf. [P-R], p. 205). If g3(J(B,σ,u1,µ1)) = g3(J(B,σ,u2,µ2)), 1 then µ− µ Nrd(B∗). 1 2 ∈ Proof. Since g3(J(B,σ,u1,µ1)) = g3(J(B,σ,u2,µ2)), we have [B] [µ1]=[B] [µ2], 1 1 ∪ ∪ which gives [B] [µ1µ2− ] = 0. Hence the algebra J(B,µ1µ2− ) is reduced ([R]). This 1 ∪ implies µ µ− Nrd(B∗). 1 2 ∈ Proposition 2.2. If f3(J(B,σ,u1,µ1)) = f3(J(B,σ,u2,µ2)), then the rank 1 her- mitian forms u and u over (B,σ) are equivalent. h 1i h 2i Proof. In view of ([BF-L]), it suffices to prove this after an odd degree base change of k.LetMbe an odd degree extension of k such that B = B M is split M ⊗k over LM = L k M . We may therefore assume that (B,σ)=(M3(L), ), where ⊗ 1 t ∗ ∗ is given by X∗ =Γ− X Γ, Γ= γ ,γ ,γ , with γ k∗. We may further assume h 1 2 3i i ∈ that γ1γ2γ3 =1.Letu1 correspond to the matrix V1 and u2 to the matrix V2 in M3(L). Then det(Vi)=µiµi,i=1,2. By Theorem 1.1, J(M (L), ,V ,µ ) (C (L3,V Γ 1,ψ ),Γ) 3 ∗ 1 1 'H3 1 1 − 1 and J(M (L), ,V ,µ ) (C (L3,V Γ 1,ψ ),Γ), 3 ∗ 2 2 'H3 2 2 − 2 1 where ψi is the trivialization of disc(ViΓ− )givenbyµi,i=1,2. We therefore have f ( (C (L3, V Γ 1,ψ ),Γ)) = f ( (C (L3, V Γ 1,ψ ),Γ)), 3 H3 1 1 − 1 3 H3 2 2 − 2 which implies that n n . So, by Corollary 1.2, C1 ' C2 tr ( 1 h ) tr ( 1 h ), L/k h i⊥ 1 ' L/k h i⊥ 2 1 where hi is the hermitian form given by the matrix ViΓ− .Thus,trL/k h1 tr h . By a theorem of Jacobson (cf. Appendix 2 of [M-H]), h h .Sothere◦ ' L/k ◦ 2 1 ' 2 1 t 1 1 t exists W GL (L) such that W V Γ W = V Γ , i.e., WV (Γ− W Γ) = V . ∈ 3 1 − 2 − 1 2 Hence V1 and V2 are hermitian equivalent in (M3(L), ), which implies that the hermitian forms u and u over (B,σ) are equivalent.∗ h 1i h 2i Proposition 2.3. Let u1, u2 be units in (B,σ) with σ(ui)=ui,n(ui)=µiµi,µi 1 ∈ L∗.Let u1 and u2 be hermitian equivalent over (B,σ) and let µ1− µ2 be a reduced norm fromh iB. Thenh i there exists w B such that ∈ u2 = wu1σ(w),µ2=n(w)µ1. Proof. We introduce an equivalence among the pairs (u, µ), where u B with ∼ ∈ σ(u)=uand n(u)=µµ, µ L∗, as follows: (u, µ) (u0,µ0) there exists w B ∈ ∼ ⇔ ∈ with u0 = wuσ(w)andµ0=n(w)µ. We show that (u ,µ ) (u ,µ ). 1 1 ∼ 2 2 Let µ = n(w )µ .Then(u,µ ) (u0,µ ), where u0 = w u σ(w ). Since u 2 1 1 1 1 ∼ 2 1 1 1 h 1i and u are hermitian equivalent, we get that u0 is hermitian equivalent to u . h 2i h i h 2i Let w0 B be such that u = w0u0σ(w0). Now ∈ 2 n(u0)=n(w1)n(w1)µ1µ1 =µ2µ2 =n(u2).

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Thus, n(w0)n(w0)=1.Letλ=n(w0). By the following lemma due to Rost, there exist w00 B such that λ = n(w00)andw00u σ(w00)=u .Thus ∈ 2 2 1 1 (u0,µ ) (u ,λ− µ ) (w00u σ(w00),n(w00)λ− µ )=(u ,µ ). 2 ∼ 2 2 ∼ 2 2 2 2 Therefore, (u ,µ ) (u ,µ ). 1 1 ∼ 2 2 Lemma 2.4 (Rost). Let L/k be a quadratic extension. Let A be a degree 3 central simple (associative) algebra over L with an involution σ of the second kind. Let x L∗ be such that xσ(x)=1. Assume that x = N(a) for some a A. Then there ∈ ∈ exists a0 A with x = N(a0) and a0σ(a0)=1. ∈ In fact, we have the following more general lemma due to Suresh, the proof of which uses the following result of Merkurjev on norm principle for unitary groups.

1 Lemma 2.5 (Merkurjev). Let x L∗.Thenx=N(vσ(v)− ) for some v A∗ if ∈ ∈ and only if there exists u A∗ with uσ(u)=1and x = N(u). ∈ Proof. (cf. Theorem 5.1.3 of [BF-P]). Lemma 2.6 (Suresh). Let L/k be a quadratic extension and A a central simple (associative) algebra over L of odd degree with an involution of the second kind. Let x L be such that xx =1.Ifxis the reduced norm of some element of A,then ∈ there exists an element u A∗ such that uσ(u)=1and N(u)=x. ∈ Proof. Let x L∗ be such that xx =1andx=N(v)forsomev A∗. Then there ∈ 1 ∈ exists y L∗ such that x = y(y)− .Sinceσis the identity on k,forλ k∗ we have ∈ 1 ∈ x = λy(λy)− . Therefore, by Lemma 2.5, it is enough to show that λy is a reduced 2 norm of some element of A∗ for some λ k∗.Let[A:L]=(2r+1) .Wehave ∈ r r 2r+1 N(vy)=N(v)(y) (since σ(y) L∗) = xr(y)2r+1 ∈ r r 2r+1 =(y(y)−)(y) =(yy)ry. r Let λ =(yy) k. Then it follows that λy is a reduced norm from A∗, and hence ∈ λy is a reduced norm from A∗. This completes the proof of lemma. Theorem 2.7. Let k be a field with characteristic different from 2 and 3. Let L/k be a quadratic extension, B a degree 3 central simple algebra over L with an involution σ of the second kind. Let ui,i=1,2,beunitsinBwith σ(ui)=ui and n(ui)=µiµi,µi L∗.ThenJ(B,σ,u1,µ1) J(B,σ,u2,µ2) if and only if there is w B with u =∈wu σ(w) and µ = n(w)µ'. ∈ 2 1 2 1 Proof. If u2 = wu1σ(w)andµ2 =n(w)µ1 then, as was remarked in the intro- duction, the map (a ,a) (a ,aw) gives an isomorphism of J(B,σ,u ,µ ) with 0 7→ o 1 1 J(B,σ,u2,µ2). Conversely, suppose J = J(B,σ,u ,µ ) J = J(B,σ,u ,µ ). Then f (J )= 1 1 1 ' 2 2 2 3 1 f3(J2)andg3(J1)=g3(J2). The theorem now follows from Propositions 2.1, 2.2 and 2.3.

Theorem 2.8. Let J = J (B,σ,u,µ),J0=J(B,σ,u0,µ0) be Albert algebras such that f (J)=f (J0)and g (J)=g (J0).ThenJ J0. 3 3 3 3 ' Proof. The proof follows from Propositions 2.1, 2.2, 2.3 and the ‘if’ part of Theo- rem 2.7.

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1 1 Remark. Theorem 2.7 asserts that the map H (k, SU(B,σ)) H (k, F4), on the Galois cohomology, corresponding to the second Tits’ construction,→ is injective.

3. A Skolem Noether type theorem In this section we prove the following “Skolem Noether” type theorem.

Theorem 3.1. Let L/k be a quadratic field extension. Let (B,σ),(B0,σ0) be degree 3 central simple algebras over L with involutions of the second kind over k.Let (B,σ), (B0,σ0) denote the 9-dimensional Jordan algebras over k associated to H H the symmetric elements in (B,σ),(B0,σ0) respectively. Suppose that (B,σ) and H (B0,σ0) are subalgebras of an Albert algebra J over k and α : (B,σ) (B0,σ0) His an isomorphism of Jordan algebras. Then α extends to an automorphismH 'H of J. For the proof we need some lemmas (which are also consequences of a more general result proved by Jacobson [J], p. 210, Theorem 11. However, we give direct proofs for the sake of completeness.) Lemma 3.2. Let L/k be a quadratic field extension. Let B be a degree 3 central simple algebra with an involution σ of the second kind. Let α : (B,σ) (B,σ) be an automorphism of Jordan algebras. Then α is the restrictionH of an isomorphism→H α :(B,σ) (B,σ) or α :(B,σ) (Bop,σ) of associative algebras with involutions. ' ' Proof. Consider the map α 1: (B,σ) k L (B,σ) k L.Themapψ: e e ⊗ H ⊗ 'H ⊗ (B,σ) k L B, given by (x, λ) xλ, is an isomorphism of Jordan algebras H ⊗ → 1 7→ over L.Letα=ψ (α 1) ψ− : B B.Thenαrestricts on (B,σ)toα.Bya theorem of Ancochea◦ ([A]),⊗ ◦α is an automorphism' or an anti-automorphismH of the algebra B. We show that α commutes with σ.Forx (B,σ),σ(x)=x,sothat e e ∈H ασe(x)=α(x)=α(x)=σα(x). e Let L = k(j) with j2 k. Then, ασ(j)=α( j)= j=σα(j). Since (B,σ) generates B as an L-algebra,∈ e it followse that α commutes− e − with σ on the wholeH of B. Hence α has the required properties.e e e e Theorem 3.3. Let B be a central simple L-algebra of degree 3 with involutions σ, σ0 e of the second kind. Let Q ,Q denote the restrictions to (B,σ) and (B,σ0), σ σ0 H H respectively, of the trace form of B. Then the involutions σ and σ0 are isomorphic

if and only if Qσ and Qσ0 are isometric. Proof. See [H-K-R-T], Proposition 4.

Lemma 3.4. Let L/k be a quadratic extension of k.Let(B,σ),(B0,σ0) be degree 3 central simple (associative) algebras over L with involutions of the second kind. If α : (B,σ) (B0,σ0) is an isomorphism of Jordan algebras, then there exists an H 'H 0op isomorphism α :(B,σ) (B0,σ0) or α :(B,σ) (B ,σ0) of associative algebras with involutions, which→ restricts to α. → Proof. The isomorphisme α extends toe a Jordan algebra isomorphism

(B,σ) L (B0,σ0) L, H ⊗k 'H ⊗k which in turn gives an isomorphism of the Jordan algebra B with B0. By a theorem 0 op of Ancochea, B is isomorphic or anti-isomorphic to B0. Replacing B0 by B ,if necessary, we may assume that B is isomorphic to B0.Inviewofthefactthatα is an isomorphism of Jordan algebras, we get that Q Q . By Theorem 3.3 and σ ' σ0

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the fact that B is isomorphic to B0 we conclude that there exists an isomorphism 0op β :(B,σ) (B0,σ0)or(B,σ) (B ,σ0) of associative algebras with involutions. ' ' 1 Thus β restricts to β : (B,σ) (B0,σ0). Consider β− α : (B,σ) (B,σ). Bye Lemma 3.2, there existsH γ :('HB,σ) (B,σ)or(B,σ) ◦(BopH,σ), which→H restricts 1 ' ' to β− e α.Letα=β γ.Thenαsatisfies the requirements in the lemma. ◦ ◦ Proof of Theorem 3.1. By Lemma 3.4 there is an isomorphism α :(B,σ) (B0,σ0) 0ope e e → or (B,σ) (B ,σ0) which restricts to α. By ([MC-2]), there are isomorphisms → φ : J(B,σ,u,µ) J, φ : J(B0,σ0,u0,µ0) eJ 1 ' 2 ' for suitable u, µ, u0,µ0, which restrict to the inclusions of (B,σ)and (B0,σ0) H H in J. We have an isomorphism J(α):J(B,σ,u,µ) J(B0,σ0,α(u),µ)givenby ' (a,a) (α(a ),α(a)). But J(B0,σ0,α(u),µ) J(B0,σ0,u0,µ0), since both are 0 7→ 0 ' isomorphic to J. By Theorem 2.2 of 2, there exists w0 B0 such that u0 = e § ∈ w0α(u)σ0(w0),µ0=N(w0)µ.Letψ:J(B0,σ0,α(u),µ) J(B0,σ0,u0,µ0)begiven ' by (a0,a0) (a0,a0w0). Then ψ restricts to the identity map on (B0,σ0). Let 7→ 1 H φ = φ2 ψ J(α) φ1− .Thenforx (B,σ), we have φ(x)=φ2ψ(α(x)) = α(x). Thus φ◦is an◦ automorphism◦ of J extending∈H α. e References

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-5, India E-mail address: parimala/sridhar/[email protected]

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