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Ring Isomorphisms ofH*-Algebras
JOURNAL OF ALGEBRA ARTICLE NO.
181, 329]346 Ž1996.
0123
Ring Isomorphisms of H U-Algebras A. R. VillenaU Departamento de Analisis Facultad de Ciencias, Uni¨ ersidad de Granada, ´ Matematico, ´ 18071 Granada, Spain
and M. Zohry † Departement de Mathematiques, Faculte´ des Sciences, Uni¨ ersite´ Abdelmalek Essaidi, ´ ´ BP 2121 Tetouan, Morocco Communicated by Georgia Benkart Received May 10, 1994
1. INTRODUCTION Associative H U -algebras were introduced by Ambrose in w1x. Since then, H -algebras have been intensively studied in the most familiar classes of nonassociative algebras and even in the general nonassociative context w7, 8x. The study of nonassociative H U -algebras can be reduced to the case of topologically simple H U -algebras by means of w8, Thm. 1x. U
THEOREM 1. E¨ ery H U-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals which are topologically simple H U-algebras themsel¨ es. Moreover, it was proved in w7x that the H U -algebra structure of topologically simple H U-algebras is in essence algebraically determined. THEOREM 2. Algebraically isomorphic H U-algebras with zero annihilator are automatically )-isomorphic, and )-isomorphisms between topologically simple H U-algebras are constant positi¨ e multiples of isometries. Accordingly, algebraically isomorphic topologically simple H U-algebras are, up to multiplication of the inner product by a suitable positi¨ e number, totally isomorphic. * E-mail: [email protected]. † Supported by the ‘‘Cooperacion Proyecto: Estruc´ interuniversitaria Marroquı-Andaluza.’’ ´ turas algebrico-topologicas no asociativas y sus aplicaciones. ´ ´ 329 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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As it is usual, by an isomorphism between given H U-algebras we mean a Žlinear. isomorphism of the underlying algebras without any reference to the H U-algebra structure, while by )-isomorphism we mean an isomorphism preserving H U-algebra involutions. A more general approach is to consider these algebras only as rings. It seems that the first work in this direction is due to Rickart w15, Thm. 3.2x, who studied ring isomorphisms of primitive real Banach algebras Žno linearity was assumed.. A well-known result of Kaplansky w10x decomposes every ring isomorphism between semisimple complex Banach algebras into a continuous linear part, a continuous conjugate-linear part, and a non-real-linear part on a finite-dimensional ideal. In this paper we prove that the H U-algebra structure of infinite-dimensional topologically simple H U-algebras is in essence determined from their ring structure and, surprisingly, the whole H U-algebra structure of any topologically simple associative H U-algebra is determined from their mere multiplicative structure. Given H U-algebras with zero annihilator A and B and a ring isomorphism F from A onto B, we state a decomposition of A into an orthogonal sum of a finite-dimensional ideal and a closed ideal on which F is continuous. Furthermore, the ‘‘continuity ideal’’ is the orthogonal sum of a closed ideal on which F is linear and a closed ideal on which F is conjugate-linear. Moreover, the ‘‘discontinuity ideal’’ decomposes into an orthogonal sum of ideals on each of which F is t-linear for a suitable automorphism t of the complex field. The result provides a nonassociative parallel of the famous result of Kaplansky. We emphasize that if A and B are associative H U-algebras with zero annihilator and finitely many one-dimensional minimal closed ideals and F is a multiplicative bijection Žit is not assumed to be additive. from A onto B, then we prove that A and B are automatically )-isomorphic. As the Kaplansky theorem was proved in w21x, we also use automatic continuity methods in our investigation. Moreover we involve the theory of central closability. This technology has been successfully exploited in the study of the continuity properties of derivations and homomorphisms on H U -structures w17, 24]27x. We recall that an H U-algebra is a nonassociative complex algebra A endowed with a conjugate-linear algebra involution ) and whose underlying vector space is a Hilbert space relative to an inner product Ž? < ? . satisfying Ž ab < c . s Ž b < aU c . s Ž a < cbU . for all a, b, c g A. We can measure goodness of the product of A by considering its annihilator, defined as AnnŽ A. s a g A : aA s Aa s 04 . We note that, by w4, Lemma 34.9x, an associative H U-algebra has zero annihilator if, and only if, it is semisimple.
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2. CENTRAL CLOSABILITY TECHNOLOGY The concept of extended centroid was introduced and developed by Martindale w12, 13x in the case of prime associative rings and later was translated to the more general frame of prime and semiprime nonassociative algebras w3, 9x. Although the main use of the extended centroid is found in the theory of structure and classification of prime algebras, central closability became unexpectedly a powerful tool in the investigation of the continuity properties of homomorphisms and derivations on H U-algebras w17, 24, 25x. We recall that a prime algebra A over a field K is said to be centrally closed if, for every nonzero ideal I of A and for every linear mapping f : I ª A with f Ž xa. s f Ž x . a and f Ž ax . s af Ž x . for all x in I and a in A, there exists l in K such that f Ž x . s l x for all x in I. EXAMPLES 1. Ž1. From w12, 13x and the standard theory of Banach algebras, it may be concluded that primitive complex Banach algebras are centrally closed. Ž2. It was proved by Rodrıguez w18x that every primitive complex ´ Jordan]Banach algebra is centrally closed. Ž3. Ara showed in w2x that prime CU -algebras are centrally closed, and this result was generalized to prime JBU -algebras in w19x. In the sequel, we shall use the following crucial result in the theory of topologically simple H U -algebras obtained in w5x. THEOREM 3.
E¨ ery topologically simple H U-algebra is centrally closed.
From the above theorem and w17, Lemma 4x, we deduce the following result. COROLLARY 1. E¨ ery dense subalgebra of a topologically simple H U-algebra is centrally closed. Let A be any nonassociative algebra. If a g A, then we denote by L a and R a the linear operators of left and right multiplication, respectively, by the element a in A. The multiplication algebra of A is defined as the subalgebra M Ž A. of LŽ A. Žthe algebra of all linear operators on A. generated by all left and right multiplication operators. It is easy to check that M Ž A. coincides with the subring of LŽ A. generated by all left and right multiplication operators. For a given subalgebra B of A, let us denote by MB Ž A. the subalgebra of M Ž A. generated by the multiplication operators L b and R b where b g B. Note that, for a normed algebra A, every element in M Ž A. is clearly a continuous linear operator on A.
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Furthermore, if B is a dense subalgebra of A, then the elements of M Ž B . are nothing but the restriction to B of those of MB Ž A.. This fact will be used in the sequel without previous comment. For a given continuous linear operator F on a Hilbert space we denote by F U its adjoint operator. Given an element a in an H U-algebra A we have LUa s L aU and RUa s R aU . Accordingly, for a dense subalgebra B of A, Ž MB Ž A..U s MBU Ž A. and M Ž A. becomes a self-adjoint subalgebra of the algebra of all continuous linear operator on the Hilbert space A. Let F be a ring homomorphism from an algebra A into an algebra B, that is an additive mapping from A into B satisfying F Ž ab. s F Ž a. F Ž b . for all a, b g A. For every a g A, we have F L a s LF Ž a. F and F R a s R F Ž a.F. Therefore the subring of those P in M Ž A. for which there exists Q in M Ž B . such that F P s QF, contains all left and right multiplication operators on A and therefore equals M Ž A.. Furthermore, if F is a ring homomorphism from A onto a dense subalgebra of a normed algebra B we obtain, for every P in M Ž A., a unique element f Ž P . in MF Ž A.Ž B . such that F P s f Ž P . F. So we can define a ring homomorphism F from M Ž A. onto MF Ž A.Ž B .. From the identity F P s f Ž P . F, it follows that if F is additionally injective, then f is a ring isomorphism from M Ž A. onto MF Ž A.Ž B .. In order to investigate the continuity properties of ring isomorphisms, we follow the traditional sliding hump procedure. To this end, we apply w25, Thm. 5x to build sequences having amazing properties, which allows us to put two powerful automatic continuity principles into action. THEOREM 4. Let B be a dense subalgebra of an infinite-dimensional topologically simple H U-algebra A. Then one of the following assertions holds: ŽC1. There exists sequences bn4 in BU and Pn4 in MB Ž A. such that PnU . . . P1U bn / 0 ŽC2.
U and Pnq1 PnU . . . P1U bn s 0
; n g N.
There exists a sequence Q n4 in MB Ž A. such that dim Q n Ž B . s 1
;n g N
Q n2 / 0
;n g N
Qn Qm s 0
if n - m.
Proof. BU is also a dense subalgebra of A, and Corollary 1 shows that it is centrally closed. Moreover, it is easy to check that BU is a topologically simple algebra. Finally, the result follows from w25, Thm. 5x, where the multiplication algebra was additionally generated by the identity operator, but this condition is unnecessary.
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3. AUTOMATIC CONTINUITY METHODS Note that, if F is an additive mapping between two Žreal or complex. vector spaces, then F is rational-linear and for rational-linear mappings between normed spaces continuity is equivalent to boundedness as for linear mappings. Moreover, we can measure the continuity of an additive mapping F between Banach spaces X and Y by considering the subset S Ž F . s y g Y : there exists x n ¬ 0 and F Ž x n . ¬ y 4 . It is easy to check that S Ž F . is a closed real-linear subspace of Y, which will be called the separating subspace for F. We observe that closed graph theorem remains true for additive mappings between Banach spaces and so the mapping F is continuous if, and only if, S Ž F . equals zero. Moreover, w21, Lemma 1.3x can be applied in order to obtain that, for any continuous linear operator G with domain Y but which may map into another Banach space Z, GF is continuous if, and only if, GŽ S Ž F .. s 0. Furthermore, if F is a ring homomorphism with dense range between normed algebras A and B, then S Ž F . is a real-linear ideal of B. For studying the continuity behaviour of a ring isomorphism acting between H U-algebras, we require two fundamental principles in the theory of automatic continuity stated in w23, 27x that remain true for additive mappings acting between Banach spaces. PROPOSITION 1 w23x. Let X be a Banach space, Sn4 a sequence of continuous linear operators from X into itself and R n4 be a sequence of continuous linear operators whose domain is a Banach space Y but which may map into others Banach spaces Yn . If F is a possibly discontinuous additi¨ e mapping from X into Y, such that R n FS1 . . . S m is continuous for m ) n, then R n FS1 . . . S n is continuous for sufficiently large n. PROPOSITION 2 w27x. Let X and Y be Banach spaces, S n4 a sequence of continuous linear operators from X into itself satisfying Sn Sm s 0 if n - m, and R n4 be a sequence of continuous linear operators from Y into itself. If F is a possibly discontinuous additi¨ e mapping from X into Y such that FS n y R n F is continuous for e¨ ery n g N, then FS n2 , and so R 2n F, is continuous for sufficiently large n.
4. RING ISOMORPHISMS OF TOPOLOGICALLY SIMPLE H U-ALGEBRAS Our main goal is to show that every ring isomorphism between infinitedimensional topologically simple H U-algebras is continuous. From this
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continuity, it will follows either the linearity or the conjugate-linearity of such isomorphism. LEMMA 1. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. If C1 is fulfilled for F Ž A., then F is continuous. Proof. For every n g N we put Pn s f Ž Sn . for a suitable Sn g M Ž A. and we define R nŽ b . s Ž b < bn . for all b g B. For every a g A and m, n g N we have R nFS1 . . . Sm a s Ž FS1 . . . Sm a < bn . s Ž f Ž S1 . . . Sm . F a < bn . s Ž f Ž S1 . . . . f Ž Sm . F a < bn . s Ž F a < PmU . . . P1U bn . , which equals zero if m ) n. From Proposition 1, it follows the continuity of the operator R nFS1 . . . S n , and hence the continuity of the functional a ¬ Ž F a < PnU . . . P1U bn . and A, for sufficiently large n. The linear subspace of B defined by J s b g B : a ¬ Ž F a < b . is continuous Ž and so real-linear . on A4 is closed by the Banach]Steinhaus theorem. It is easy to check that J is an ideal of B which contains the nonzero element PnU . . . P1U bn for sufficiently large n. Consequently J equals B by topological simplicity. The continuity of every functional a ¬ Ž F a < b . Ž b g B . shows that Ž S Ž F .< B . s 0 which implies that SŽ F . s 0 and hence the continuity of F. LEMMA 2. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. Then ker F s 0. Proof. Suppose the lemma were false. Then ker F q i ker F is a closed nonzero ideal in A and so ker F q i ker Fs A. Since Žker F q i ker F . A s Ž ker F . A q Ž ker F .Ž iA . ; ker F , it follow s th at A 2 s ker F q i ker FA ; ker F. Since A is the closure of the set Ý nks 1 a k bk : a k , bk g A, k s 1, . . . , n, n g N4 w8, Prop. 2Žviii.x, it may be concluded that ker Fs A. Let ker f denote the closure in M Ž A. of ker f . If a g A, then a s lim a n for a suitable sequence a n4 in ker F. Clearly L a s lim L a n , R a s lim R a n and further L a n , R a n g ker f ; n g N. Hence L a , R a g ker f. Since ker f is obviously a subalgebra of M Ž A. we conclude that ker fs M Ž A.. If F Ž A. fulfills ŽC1., then the preceding lemma shows that F is continuous and so ker F is closed. From what has already been proved, it follows that ker F s A and therefore F s 0, a contradiction.
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On the other hand, if F Ž A. satisfies ŽC2., then we have an element Q g MF Ž A.Ž B . with dim QŽ B . s 1 and Q 2 / 0. There are b1 , b 2 g B such that QŽ b . s Ž b < b1 . b 2 ; b g B. Furthermore Ž b 2 < b1 . / 0, since Q 2 / 0. Let S s Ž b 2 < b1 .y1 Q g MF Ž A.Ž B . and note that S / 0 and S 2 s S. There exists P g ker f such that f Ž P . s S and we take R g ker f such that 5 R y P 5 - 1. Consequently Ž IA y P q R . F s IA for a suitable invertible continuous linear operator F from A onto itself. Hence, we have F s F Ž IA y P q R . F s Ž F y F P q F R . F s Ž F y f Ž P . F q f Ž R . F . F s Ž F y SF . F s Ž IB y S . F F and so IB y S is a densely valued continuous linear operator from B onto itself. Since SŽ IB y S . s 0, we conclude that S s 0, a contradiction. THEOREM 5. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. Then F is continuous and either a linear or a conjugate-linear isomorphism from A onto B. Proof. On account of Lemma 1, if F Ž A. fulfills ŽC1., then F is continuous. On the other hand, if F Ž A. fulfills ŽC2., then for each n g N we define Sn s fy1 Ž Q n .. Moreover, we set R n s Q n . According to Proposition 2, the operator Q n2 F is continuous and therefore S Ž F . ; kerŽ Q n2 ., for sufficiently large n. The ideal S Ž F . q i S Ž F . is contained in the closed linear subspace kerŽ Q n2 . for sufficiently large n. The topological simplicity of B implies that S Ž F . q i S Ž F . s 0 and so S Ž F . s 0, which shows the continuity of F. For studying the linearity of F, we follow the pattern established in w21, Thm. 7.6x. Let c 1 and c 2 be the operations of multiplication by the complex scalar i in A and B respectively. Then Ž Fy1c 2 F . 2 s c 12 s yIA and it is satisfied that Ž c 1Fy1c 2 F .Ž a. b s Ž Fy1c 2 Fc 1 .Ž a. b and bŽ c 1Fy1c 2 F .Ž a. s bŽ Fy1c 2 Fc 1 .Ž a. for all a, b g A. Thus Ž c 1Fy1c 2 F .Ž a. y Ž Fy1c 2 Fc 1 .Ž a. g AnnŽ A. s 0 for every a g A and so Fy1c 2 F and c 1 commute. If we define w s c 1Fy1c 2 F, then w is a continuous real-linear operator on A with w 2 s IA . Furthermore, w Ž ab. s w Ž a. b s a w Ž b . for all a, b g A. In particular, w satisfies w Ž iab. s iaw Ž b . s i w Ž ab. which shows that w is linear on A2 . Since w is continuous, w8, Prop. 2Žviii.x shows that w is a continuous linear operator on A. Let Aqs Ž w y IA . A and Ays Ž w q IA . A which are closed ideals of A, and A s Aq[ Ay. From the topological simplicity of A it follows that either A s Aq or A s Ay. Moreover, F is linear on Aq and conjugate-linear on Ay.
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Finally, to prove that F maps A onto B, we apply w7, Thm. 3.1x in the linear case. On the other hand, if F is a conjugate-linear operator, then we consider the topologically simple H U-algebra BX , the scalar product and inner product in BX being l ? a s l a and ² a, b : s Ž b < a. Ž l g C, a, b g B ., respectively, the sum, product, and involution being the same as those in B. F is a continuous linear homomorphism densely valued into BX . From w7, Thm. 3.1x, we deduce that F is onto, which completes the proof. EXAMPLES 2. Ž1. For any topologically simple associative H U -algebra A the symmetrized algebra Aq becomes a topologically simple Jordan H U -algebra Žsee w8x.. Every Jordan ring isomorphism F between two infinite-dimensional topologically simple associative H U-algebras A and B becomes a ring isomorphism from Aq onto Bq. From the above theorem we deduce that F is either a linear or a conjugate-linear continuous operator. Ž2. Any infinite-dimensional topologically simple associative H U -algebra A gives rise to a topologically simple Lie H U-algebra Ay, the multiplication in Ay being the usual Lie multiplication w x, y x s 12 Ž xy y yx ., the involution and inner product being the same as those in A Žsee w6x.. Every Lie ring isomorphism F between two infinite-dimensional topologically simple associative H U-algebras A and B becomes a ring isomorphism from Ay onto By and therefore is either a linear or a conjugatelinear continuous operator. Ž3. Let H1 and H2 be infinite-dimensional complex Hilbert spaces and J1 and J 2 conjugations on H1 and H2 , respectively, that is Jk is a conjugate-linear isometry from Hk onto itself satisfying Jk2 s IH k for k s 1, 2. Assume that t is an automorphism of the complex field and F any t-linear bijective map from H1 onto H2 for which the identity
Ž Fx < J2 Fy . s t Ž x < J1 y . holds for all x, y g H1. Then the Hilbert spaces A k s C [ Hk , with the products Ž l q x k .Ž m q y k . s lm q Ž x k < Jk y k . q l y k q m x k and involutions Ž l q x k .U s l q Jk Ž x k ., are infinite-dimensional topologically simple Jordan H U -algebras for k s 1, 2 Žsee w8x. and the map F Ž l, x . s Žt Ž l., Fx . becomes a ring isomorphism from A1 onto A 2 . On account of the above theorem, it follows that F is continuous and consequently F is continuous and t is either the identity or the conjugation on C. COROLLARY 2. Let A and B be infinite-dimensional topologically simple H U-algebras and F be any ring isomorphism from A onto B. Then F is either a linear or a conjugate-linear continuous isomorphism from A onto B and
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pro¨ ides either a linear or a conjugate-linear )-isomorphism C from A onto B. Furthermore there exists a positi¨ e constant r such that r C is isometric. Proof. From the preceding theorem, it follows that F is either a linear or a conjugate-linear continuous isomorphism from A onto B. If F is linear, then the remainder of the result follows from Theorem 2. If F is a conjugate-linear isomorphism we consider the topologically simple H U-algebra BX defined as at the end of the proof of Theorem 5. F is a linear isomorphism from A onto BX and consequently Theorem 2 assures the existence of a )-isomorphism C from A onto BX and a positive number r such that r ? C s r C is isometric. Therefore C is a conjugate-linear )-isomorphism from A onto B. Since the norm associated to the new inner product on B coincides with the original one, it follows that r C is isometric. Obviously in the finite-dimensional setting real-linearity and continuity are equivalent. Surprisingly, a discontinuous ring isomorphism between finite-dimensional topologically simple H U-algebras may be constructed. We give below an example of this situation. EXAMPLE 3. Let m be the automorphism from the subfield QŽ'2 . of C onto itself given by m Ž r q s'2 . s r y s'2 Ž r, s g Q.. Now we consider the set of all automorphisms of some subfield of C extending m , which is inductively ordered by extension. Let t : K ª K be a maximal element. We claim that t is an automorphism of the whole complex field. Otherwise there exists a complex number x with x f K. We apply w11, Thm. II.67x to obtain that K is algebraically closed and so x is transcendent over K. Therefore the automorphism t can be extended to an automorphism of KŽ x . and this contradiction proves our claim. Consequently t is a nonreal-linear Žand so discontinuous. automorphism of the complex field. Next, we show an intimate connection between Lebesgue measurability, continuity and linearity for additive maps. LEMMA 3. Let F be an additi¨ e mapping from R into itself. Then the following assertions are equi¨ alent: Ž1. Ž2. Ž3.
F is linear. F is continuous. F is measurable.
Proof. We only need to show that the third assertion implies the first one. Since R s D `ns 1 Fy1 Žwyn, n x., there exists n g N such that lŽ Fy1 Žwyn, n x.. ) 0, where l denotes the Lebesgue measure on R. By w22, Thm. 6.67x there exists d ) 0 such that x y d , d w; Fy1 Žwyn, n x. y
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Fy1 Žwyn, n x. ; Fy1 Žwy2 n, 2 n x., since F is additive. Consequently F Žx y d , d w. ; wy2 n, 2 n x and therefore F is bounded on a neighbourhood of zero, which shows that F is continuous on R. LEMMA 4. Let F be an additi¨ e mapping from a finite-dimensional Banach space X into a Banach space Y. Then the following assertions are equi¨ alent: Ž1. F is real-linear. Ž2. For e¨ ery element x in X and e¨ ery continuous linear functional f on Y, the mapping l ¬ R f Ž F Ž l x .. from R into itself is measurable. Proof. If Ž1. is fulfilled, then F is continuous and so the second assertion is obviously satisfied. According to the above lemma, if assertion Ž2. is satisfied, then R fF Ž l x . s R f Ž l Fx . for every x g X, every continuous linear functional f on Y, and every l g R. From the Hahn]Banach theorem, it follows that F is real-linear. PROPOSITION 3. Let A and B be topologically simple H U-algebras with A finite-dimensional and let F be a ring homomorphism from A into B. Then the following assertions are equi¨ alent: Ž1. F is real-linear and accordingly continuous. Ž2. There is a / 0 in A such that, for all b g B, the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable. Ž3. For all a g A and b g B the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable. Proof. On account of the preceding lemma, it suffices to prove that Ž3. follows from Ž2.. To this end we consider the set I of those a g A for which the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable for every b g B. It is a simple matter to show that I is a nonzero ideal of A. Since A is finite-dimensional and topologically simple, it follows that I s A, which is the desired conclusion. Remark 1. It is known that any proof of the existence of nonmeasurable subsets of R requires, at some stage, uncountably many arbitrary choices of elements of R. The above connection between linearity and measurability shows that the use of Zorn lemma is unavoidable to construct discontinuous ring isomorphisms between finite-dimensional topologically simple H U-algebras. One can make the following somewhat vague statement. Any ring isomorphism whose values may be designated by elementary analytical constructions is measurable and consequently continuous. Loosely speaking: if you have a discontinuous ring isomorphism
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between topologically simple H U-algebras, then you cannot tell me which values it assumes, although you might be able to tell some of them. Although ring isomorphisms between finite-dimensional topologically simple H U-algebras may be non real-linear, next we prove that they satisfy a sort of linearity. THEOREM 6. Let F be a ring isomorphism between two centrally closed prime algebras A and B o¨ er a field K. Then there exists an automorphism t of K such that F Ž l a. s t Ž l. F a for all l g K and a g A. Proof. Let fix a nonzero element a in A. For every l g K, we claim that there exists taŽ l. g K such that F Ž l a. s taŽ l. F a. Assume that QŽ F a. s 0 with Q g M Ž B . and write P s fy1 Ž Q .. We have F Ž Pa. s QŽ F a. s 0 which shows that Pa s 0 and therefore 0 s F Ž l Pa. s F Ž P Ž l a.. s QŽ F Ž l a... Consequently, QŽ F a. ¬ QŽ F Ž l a.. is a well-defined linear mapping Žsay f . from the nonzero ideal I s M Ž B .Ž F a. into B satisfying f Ž by . s bf Ž y . and f Ž yb . s f Ž y . b for all y g I and b g B. Since B is centrally closed, there exists taŽ l. g K such that QŽ F Ž l a. y taŽ l. F a. s 0 for every Q g M Ž B . and so F Ž l a. s taŽ l. F a. Now we prove that the scalar taŽ l. is independent of which nonzero element a we choose to define it. Let b be a nonzero element in A. For all Q, QX g M Ž B ., we have Q Ž F Ž l b . . QX Ž F a . s F Ž fy1 Ž Q . Ž l b . . F Ž fy1 Ž QX . Ž a . . s F Ž fy1 Ž Q . Ž l b . fy1 Ž QX . Ž a . . s F Ž fy1 Ž Q . Ž b . fy1 Ž QX . Ž l a . . s F Ž fy1 Ž Q . Ž b . . F Ž fy1 Ž QX . Ž l a . . s Q Ž F b . QX Ž F Ž l a . . s Q Ž F b . QX Ž ta Ž l . F a . s Q Ž ta Ž l . F b . QX Ž F a . . The ideals M Ž B .ŽtaŽ l. F b y F Ž l b .. and M Ž B .Ž F a. have zero product. By primeness M Ž B .ŽtaŽ l. F b y F Ž l b .. equals zero, since M Ž B .Ž F a. / 0. Thus taŽ l. F b y F Ž l b . s 0 and therefore taŽ l. s t b Ž l.. We will denote this scalar by tF Ž l.. Now it is easy to check that tF Ž l q m . s tF Ž l. q tF Ž m . and tF Ž lm . s tF Ž l.tF Ž m . for all l, m g K and tF y1 tF s tFtF y 1 s IK , which shows that tF is an automorphism of the field K. Remark 2. It is easy to check that if in the preceding theorem A and B are in addition normed algebras and F is continuous, then so is t .
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Remark 3. From the preceding theorem, we deduce that ring isomorphisms between centrally closed prime algebras preserve the algebraic dimension. EXAMPLE 4. Every ring isomorphism between any of the algebras showed in Examples 1 is automatically t-linear for a suitable automorphism t of C. Since topologically simple H U-algebras are centrally closed prime algebras, we get the following. COROLLARY 3. Let F be a ring isomorphism between two topologically simple H U-algebras A and B. Then F is t-linear for a suitable automorphism t of the complex field. EXAMPLE 5. Schue proved in w20x that finite-dimensional simple Lie complex algebra can be structured as H U-algebras. On account of the preceding corollary, we deduce that every ring isomorphism between such algebras is automatically t-linear for a suitable automorphism t of C. COROLLARY 4. Ring isomorphic topologically simple associati¨ e H U-algebras are automatically )-isomorphic. Proof. Let A and B topologically simple associative H U-algebras and F be a ring isomorphism from A onto B. By the preceding result, F is t-linear for a suitable automorphism t of the complex field. By w16, Thm. 4.10.32x, A is )-isomorphic to a ‘‘full matrix algebra’’ M G . The mapping Ž lgn .g , n g G ¬ Žty1 Ž lgn ..g , n g G is a ty1 -linear automorphism of M G Žnote that, if G is infinite, then t is either the identity or the conjugation on the complex field. which gives a ty1 -linear automorphism C of A. FC is a linear isomorphism from A onto B which provides, by Theorem 2, a )-isomorphism from A onto B. Note that every topologically simple associative H U-algebra is prime and, if it has dimension greater than one, then it contains nontrivial idempotents. Consequently, w14, Corollaryx now yields the following. LEMMA 5. E¨ ery multiplicati¨ e bijecti¨ e mapping from a topologically simple associati¨ e H U-algebra with dimension greater than one onto an arbitrary associati¨ e ring is automatically additi¨ e. EXAMPLE 6. It should be noted that the conclusion of the above-established result fails if the H U-algebra is one-dimensional. Indeed, the inversion mapping l ¬ ly1 Žwith 0 ¬ 0. from C into itself is multiplicative and nonadditive. In spite of the preceding example, we get the following multiplicative characterization of the topologically simple associative H U-algebras.
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THEOREM 7. Multiplicati¨ ely isomorphic topologically simple associati¨ e H U-algebras are automatically )-isomorphic. Proof. Let A and B topologically simple associative H U-algebras and F a multiplicative bijective mapping from A onto B. If A and B have dimension one, then both of them are )-isomorphic to the complex field and consequently the conclusion in the theorem is true. If either A or B has dimension greater than one, then we conclude from Lemma 5 that F is a ring isomorphism, hence that A and B are )-isomorphic by Corollary 4. Remark 4. It is worth pointing out that multiplicative bijective mapping between topologically simple associative H U-algebras preserves the algebraic dimension. Questions 1. The above established results lead naturally to the following questions: Ž1. Are ring isomorphic topologically simple Žnonassociative. H U -algebras automatically )-isomorphic? Ž2. Are multiplicatively isomorphic topologically simple Žnonassociative. H U -algebras automatically )-isomorphic?
5. RING ISOMORPHISMS OF H U-ALGEBRAS WITH ZERO ANNIHILATOR The study of ring isomorphisms between H U-algebras with zero annihilator can be reduced to the case of ring isomorphisms between topologically simple H U-algebras. LEMMA 6. Let A and B be nonassociati¨ e rings and F be a multiplicati¨ e mapping from A onto B. Then F Ž0. s 0. Proof. Put 0 s F Ž a. for a suitable a g A. We have F Ž0. s F Ž a0. s F Ž a. F Ž0. s 0F Ž0. s 0. LEMMA 7. Let A and B be H U-algebras with zero annihilator and F be a multiplicati¨ e bijecti¨ e mapping from A onto B. If I is a closed ideal in A, then F Ž I . is an ideal of B. Proof. We only need to show that F Ž I . q F Ž I . ; F Ž I . and lF Ž I . ; F Ž I . for every l g C.
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Since F is surjective there exists a subset J of A such that F Ž J . s F Ž I . q F Ž I .. On account of the preceding lemma, we have F Ž JI H . s F Ž J . F Ž I H . s Ž F Ž I . q F Ž I . . F Ž I H . ; F Ž II H . q F Ž II H . s 0 and hence that JI H s 0. From w8, Prop. 2Žvii.x, it follows that J ; I and therefore F Ž I . q F Ž I . ; F Ž I .. The set lF Ž I . may be handled in much the same way. Indeed, let J ; A satisfying F Ž J . s lF Ž I .. Since F Ž JI H . s F Ž J . F Ž I H . s lF Ž I . F Ž I H . s lF Ž II H . s 0, we have JI H s 0 and consequently J ; I. Hence lF Ž I . ; F Ž I .. LEMMA 8. Let A and B be H U-algebras with zero annihilator and F be a multiplicati¨ e bijecti¨ e mapping from A onto B. If I is a minimal closed ideal in A, then F Ž I . is a minimal closed ideal of B. Proof. If F Ž I . J s 0 for every minimal closed ideal J of B, then by Theorem 1 it follows that F Ž I . B s 0. Therefore F Ž I . ; AnnŽ B . s 0 and so I s 0 which leads to a contradiction. Consequently there exists a minimal closed ideal J of B for which F Ž I . J / 0, equivalently IFy1 Ž J . / 0. We claim that F Ž I . ; J. Since J is a topologically simple H U -algebra Žsee Theorem 1. and F Ž I . l J is a nonzero closed ideal, we conclude that F Ž I . l J s J and therefore J ; F Ž I . . Decompose F Ž I . into an orthogonal sum J [ J X with J X ; J H . Since Fy1 Ž J . Fy1 Ž J X . s Fy1 Ž JJ X . s 0, we have
Ž Fy1 Ž J . l I .Ž Fy1 Ž J X . l I . ; Fy1 Ž J . Fy1 Ž J X . ; Fy1 Ž J . Fy1 Ž J X . s 0, and the primeness of I shows that Fy1 Ž J X . l I s 0 Žnote that 0 / IFy1 Ž J . ; Fy1 Ž J . l I .. Consequently, 0 s F Ž Fy1 Ž J X . I . s J X F Ž I . and J XF Ž I . s 0. Hence 0 s J XF Ž I . s J X Ž J q J X . s J X J X , and the semiprimeness of B shows that J X s 0. Thus F Ž I . s J, and the claim follows. The same proof works for the nonzero closed ideal Fy1 Ž J . l I of I, which gives Fy1 Ž J . ; I. Consequently, F Ž I . s J as required. The above-established result shows that every ring isomorphism between two given H U-algebras with zero annihilator provide ring isomorphisms
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between topologically simple H U-algebras. The preceding section works in this situation, and therefore we can formulate the following result. THEOREM 8. Let A and B be H U-algebras with zero annihilator and F be a ring isomorphism from A onto B. Then A and B decompose into orthogonal sums of closed ideals A s Aq[ Ay[ A1 [ ??? [ A n and B s Bq[ By[ B1 [ ??? [ Bn , with A1 , . . . , A n and B1 , . . . , Bn finite-dimensional, in such a way that the restriction FAq is a continuous linear isomorphism from Aq onto Bq, the restriction FAy is a continuous conjugate-linear isomorphism from Ay onto By, and, for k s 1, . . . , n, the restriction FA k is a t k-linear isomorphism from A k onto Bk for a suitable automorphism t k of the complex field. Proof. Let I be a minimal closed ideal of A. The restriction FI is a ring isomorphism from I onto the minimal closed ideal F Ž I . of B. On account of Theorem 1, I and F Ž I . are topologically simple H U -algebras. If in addition I has infinite dimension, then Corollary 2 shows that F is either a linear or a conjugate-linear continuous isomorphism from I onto F Ž I .. We claim that the set of those finite-dimensional minimal closed ideals on which F is discontinuous is finite. Suppose the claim were false. Then there exists a sequence In4 of pairwise different finite-dimensional minimal closed ideals in A on each of which F is discontinuous. For every n g N, MI nŽ A. acts topologically irreducibly on In and consequently it acts irreducibly on In , since In is finite-dimensional. Therefore, for every n g N, there exists Sn g MI nŽ A. with S n2 / 0. Since In Im s 0 if n / m, it follows that Sn Sm s 0 if n / m. Proposition 2 now shows that the operator f Ž Sn2 . FI n is continuous, and so S Ž FI n . ; ker f Ž Sn2 ., for sufficiently large n. Therefore the ideal S Ž FI n . q i S Ž FI n . is contained in the closed linear subspace ker f Ž Sn2 ., for sufficiently large n. Since f Ž S n2 . / 0 and F Ž In . is topologically simple, we conclude that S Ž FI n . s 0 and hence that FI n is continuous for sufficiently large n. This contradicts our assumption. Let us denote by Iq the set of all minimal closed ideals of A on which F is a continuous linear map, by Iy the set of those on which F is a continuous conjugate-linear map, and by A1 , . . . , A n the finitely many exceptional minimal closed ideals. Furthermore, we write Jqs F Ž I . : I g Iq 4 , Jys F Ž I . : I g Iy 4 , and B1 s F Ž A1 ., . . . , Bn s F Ž Bn .. Finally, we define As and Bs as the closure of the orthogonal sum of the ideals in Is and Js , respectively, for s g q, y4 . let A C s Aq[ Ay and BC s Bq[ By. Since F Ž A C . Bk s F Ž A C . F Ž A k . s F Ž A C A k . s 0 for k s 1, . . . , n, w8, Prop. 2Žvii.x shows that F Ž A C . ; F nks1 BkH s BC . Consequently S Ž FA C . ; BC . Let b g S Ž FA C .. Then b s lim F a n for a suitable sequence
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a n4 in A C converging to zero. For every minimal closed ideal I of A on which F is continuous and every a g I, we have bF a s lim F a nF a s lim F Ž a n a . s 0. Hence S Ž FA C . F Ž I . s 0. From this we deduce that S Ž FA C . BC s 0 and w8, Prop. 2Žvii.x now leads to S Ž FA C . ; BCH . From what has already been proved, it may be concluded that S Ž FA C . s 0, which shows that F is continuous on A C . From this we deduce that F Ž Aq . ; Bq and F Ž Ay . ; By. We now apply the preceding argument again, with F replaced by Fy1 , to obtain that Fy1 Ž Bq . ; Aq and Fy1 Ž By . ; Ay, which completes the proof. COROLLARY 5. Ring isomorphic associati¨ e H U-algebras with zero annihilator are automatically )-isomorphic. Proof. Let F be a ring isomorphism between two given associative H U-algebras A and B with zero annihilator, and consider the decomposition given in the preceding theorem. By Theorem 2, the H U-algebras Aq and Bq are )-isomorphic. By Corollary 4, every H U-algebra A k is )-isomorphic to Bk for k s 1, . . . , n. It is a simple matter to construct a conjugate-linear automorphism C of Ay Žtake into account the proof of Corollary 4.. FAy C is a linear isomorphism from Ay onto By. Consequently Ay and By are )-isomorphic, which completes the proof. LEMMA 9. Let A be an associati¨ e H U-algebra with zero annihilator which has no one-dimensional minimal closed ideals and let F be a multiplicati¨ e bijecti¨ e mapping from A onto an arbitrary associati¨ e ring B. Then F is automatically additi¨ e. Proof. Let us denote by I the set of all minimal closed ideals of A. Every I g I is a topologically simple associative H U-algebra with dimension greater than one and therefore it contains a nontrivial idempotent e I . We shall have established the lemma if we prove that A satisfies the conditions in the Martindale Theorem w14x for the set e I : I g I 4 of idempotents. If e I Aa s 0 for each I g I , then Ž Ae I A. a s 0 and consequently Ae I Aa s 0 for each I g I . Since Ae I As I, we conclude that Ia s 0 for each I g I and therefore Aa s 0 ŽTheorem 1.. Since A has zero annihilator we conclude that a s 0. If e I ae I AŽ1 y e I . s 0 for a given I g I , then Ž e i ae I . I Ž1 y e I . I s 0 and so Ž e I ae I .I Ž 1 y e I . Is 0. Since I Ž 1 y e I . Is I, we have Ž e I aeI . I s 0, and hence e I aeI g AnnŽ I . s 0. THEOREM 9. Let A and B be H U-algebras with zero annihilator and finitely many one-dimensional minimal closed ideals. If there exists a multiplicati¨ e bijecti¨ e mapping F from A onto B, then A and B are automatically )-isomorphic.
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Proof. Let us denote by I1 and J1 the set of all one-dimensional minimal closed ideals of A and B, respectively, and let I2 and J2 denote the sets of those which have dimension greater than one. According to Lemma 8 and Remark 4, we have
F Ž I . : I g I1 4 s J : J g J1 4
and
F Ž I . : I g I2 4 s J : J g J2 4 .
We write A k and Bk Ž k s 1, 2. for the closure in A and B, respectively, of the orthogonal sum of the elements of Ik and Jk , respectively. For every ideal J g J1 , we put J s F Ž I . for a suitable I g I1 and therefore F Ž A 2 . J s F Ž A 2 . F Ž I . s F Ž A 2 I . s 0. Hence F Ž A 2 . B1 s 0 and w8, Prop. 2Žvii.x shows that F Ž A 2 . ; B1H s B2 . The same reasoning applies to Fy1 gives Fy1 Ž B2 . ; A 2 and consequently F Ž A 2 . s B2 . According to the preceding lemma, F is a ring isomorphism from A 2 onto B2 and Corollary 5 shows that A 2 and B2 are )-isomorphic. Since A1 and B1 are obviously )-isomorphic, it may be concluded that A and B are )-isomorphic. Question 2. Are multiplicatively isomorphic associative H U-algebras with zero annihilator and infinitely many one-dimensional minimal closed ideals )-isomorphic?
REFERENCES 1. W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 Ž1945., 364]386. 2. P. Ara, Extended centroid of CU -algebras, Arch. Math. 54 Ž1990., 358]364. 3. W. E. Baxter and W. S. Martindale III, Central closure of semiprime nonassociative rings, Comm. Algebra 7 Ž1979., 1103]1132. 4. F. F. Bonsall and J. Duncan, ‘‘Complete Normed Algebras,’’ Springer-Verlag, Berlin 1973. 5. M. Cabrera and A. Rodrıguez, Extended centroid and central closure of semi-prime ´ normed algebras: A first approach, Comm. Algebra 18 Ž1990., 2293]2326. 6. J. A. Cuenca, A. Garcia, and C. Martin, Structure theory for LU-algebras, Math. Proc. Cambridge Philos. Soc. 107 Ž1990., 361]365. 7. J. A. Cuenca and A. Rodrıguez, Isomorphisms of H U-algebras, Math. Proc. Cambridge ´ Philos. Soc. 97 Ž1985., 93]99. 8. J. A. Cuenca and A. Rodrıguez, Structure theory for noncommutative Jordan H U-alge´ bras, J. Algebra 106 Ž1987., 1]14. 9. T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime Nonassociative algebra, Pacific J. Math. 60 Ž1975., 59]63. 10. I. Kaplansky, Ring isomorphisms of Banach algebras, Canad. J. Math. 6 Ž1954. 374]381. 11. I. Kaplansky, ‘‘Fields and Rings,’’ Chicago Lectures in Mathematics, Univ. of Chicago Press, 1972.
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12. W. S. Martindale III, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 Ž1969., 437]455. 13. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 Ž1969., 574]584. 14. W. S. Martindale III, When are multiplicative mappings additive? Proc. Amer. Math. Soc. 21 Ž1969., 695]698. 15. C. E. Rickart, Representations of certain Banach algebras on Hilbert space, Duke Math. J. 18 Ž1951., 27]39. 16. C. E. Rickart, ‘‘General Theory of Banach Algebras,’’ Van Nostrand, New York, 1960. 17. A. Rodrıguez, Continuity of densely valued homomorphisms into H U-algebras, Quart. J. ´ Math. Oxford 46 Ž1995., 107]118. 18. A. Rodrıguez, Primitive nonassociative normed algebras and extended centroid, in ´ ‘‘Nonassociative Algebraic Models’’ ŽS. Gonzalez and H. Ch. Myung, Eds.., pp. 233]243, Nova Science Publishers, New York, 1992. 19. A. Rodrıguez and A. R. Villena, Centroid and extended centroid of JBU-algebras, in ´ ‘‘Nonassociative Algebraic Models’’ ŽS. Gonzalez and H. Ch. Myung, Eds.., pp. 223]232, Nova Science Publishers, New York, 1992. 20. J. R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc. 95 Ž1960., 69]80. 21. A. M. Sinclair, ‘‘Automatic Continuity of Linear Operators,’’ London Math. Soc. Lecture Note Series 21, Cambridge Univ. Press, Cambridge, UK, 1976. 22. K. R. Stromberg, ‘‘An Introduction to Classical Real Analysis,’’ Wadsworth International Group, Belmont, CA. 23. M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 Ž1993., 139]152. 24. A. R. Villena, Continuity of derivations on H U-algebras, Proc. Amer. Math. Soc. 122 Ž1994., 821]826. 25. A. R. Villena, Stochastic continuity of random derivations on H U-algebras, Proc. Amer. Math. Soc. 123 Ž1995., 185]196. 26. A. R. Villena, B. Zalar, and M. Zohry, Continuity of derivation pairs on Hilbert triple systems, Boll. Un. Mat. Ital. B Ž7 . 9 Ž1995., 459]477. 27. A. R. Villena and M. Zohry, Continuity of homomorphism pairs into H U-triple systems, Proc. Ž Math. Sci.. I. A.Sc. 106 Ž1996., 1]12.
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Ring Isomorphisms of H U-Algebras A. R. VillenaU Departamento de Analisis Facultad de Ciencias, Uni¨ ersidad de Granada, ´ Matematico, ´ 18071 Granada, Spain
and M. Zohry † Departement de Mathematiques, Faculte´ des Sciences, Uni¨ ersite´ Abdelmalek Essaidi, ´ ´ BP 2121 Tetouan, Morocco Communicated by Georgia Benkart Received May 10, 1994
1. INTRODUCTION Associative H U -algebras were introduced by Ambrose in w1x. Since then, H -algebras have been intensively studied in the most familiar classes of nonassociative algebras and even in the general nonassociative context w7, 8x. The study of nonassociative H U -algebras can be reduced to the case of topologically simple H U -algebras by means of w8, Thm. 1x. U
THEOREM 1. E¨ ery H U-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals which are topologically simple H U-algebras themsel¨ es. Moreover, it was proved in w7x that the H U -algebra structure of topologically simple H U-algebras is in essence algebraically determined. THEOREM 2. Algebraically isomorphic H U-algebras with zero annihilator are automatically )-isomorphic, and )-isomorphisms between topologically simple H U-algebras are constant positi¨ e multiples of isometries. Accordingly, algebraically isomorphic topologically simple H U-algebras are, up to multiplication of the inner product by a suitable positi¨ e number, totally isomorphic. * E-mail: [email protected]. † Supported by the ‘‘Cooperacion Proyecto: Estruc´ interuniversitaria Marroquı-Andaluza.’’ ´ turas algebrico-topologicas no asociativas y sus aplicaciones. ´ ´ 329 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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As it is usual, by an isomorphism between given H U-algebras we mean a Žlinear. isomorphism of the underlying algebras without any reference to the H U-algebra structure, while by )-isomorphism we mean an isomorphism preserving H U-algebra involutions. A more general approach is to consider these algebras only as rings. It seems that the first work in this direction is due to Rickart w15, Thm. 3.2x, who studied ring isomorphisms of primitive real Banach algebras Žno linearity was assumed.. A well-known result of Kaplansky w10x decomposes every ring isomorphism between semisimple complex Banach algebras into a continuous linear part, a continuous conjugate-linear part, and a non-real-linear part on a finite-dimensional ideal. In this paper we prove that the H U-algebra structure of infinite-dimensional topologically simple H U-algebras is in essence determined from their ring structure and, surprisingly, the whole H U-algebra structure of any topologically simple associative H U-algebra is determined from their mere multiplicative structure. Given H U-algebras with zero annihilator A and B and a ring isomorphism F from A onto B, we state a decomposition of A into an orthogonal sum of a finite-dimensional ideal and a closed ideal on which F is continuous. Furthermore, the ‘‘continuity ideal’’ is the orthogonal sum of a closed ideal on which F is linear and a closed ideal on which F is conjugate-linear. Moreover, the ‘‘discontinuity ideal’’ decomposes into an orthogonal sum of ideals on each of which F is t-linear for a suitable automorphism t of the complex field. The result provides a nonassociative parallel of the famous result of Kaplansky. We emphasize that if A and B are associative H U-algebras with zero annihilator and finitely many one-dimensional minimal closed ideals and F is a multiplicative bijection Žit is not assumed to be additive. from A onto B, then we prove that A and B are automatically )-isomorphic. As the Kaplansky theorem was proved in w21x, we also use automatic continuity methods in our investigation. Moreover we involve the theory of central closability. This technology has been successfully exploited in the study of the continuity properties of derivations and homomorphisms on H U -structures w17, 24]27x. We recall that an H U-algebra is a nonassociative complex algebra A endowed with a conjugate-linear algebra involution ) and whose underlying vector space is a Hilbert space relative to an inner product Ž? < ? . satisfying Ž ab < c . s Ž b < aU c . s Ž a < cbU . for all a, b, c g A. We can measure goodness of the product of A by considering its annihilator, defined as AnnŽ A. s a g A : aA s Aa s 04 . We note that, by w4, Lemma 34.9x, an associative H U-algebra has zero annihilator if, and only if, it is semisimple.
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2. CENTRAL CLOSABILITY TECHNOLOGY The concept of extended centroid was introduced and developed by Martindale w12, 13x in the case of prime associative rings and later was translated to the more general frame of prime and semiprime nonassociative algebras w3, 9x. Although the main use of the extended centroid is found in the theory of structure and classification of prime algebras, central closability became unexpectedly a powerful tool in the investigation of the continuity properties of homomorphisms and derivations on H U-algebras w17, 24, 25x. We recall that a prime algebra A over a field K is said to be centrally closed if, for every nonzero ideal I of A and for every linear mapping f : I ª A with f Ž xa. s f Ž x . a and f Ž ax . s af Ž x . for all x in I and a in A, there exists l in K such that f Ž x . s l x for all x in I. EXAMPLES 1. Ž1. From w12, 13x and the standard theory of Banach algebras, it may be concluded that primitive complex Banach algebras are centrally closed. Ž2. It was proved by Rodrıguez w18x that every primitive complex ´ Jordan]Banach algebra is centrally closed. Ž3. Ara showed in w2x that prime CU -algebras are centrally closed, and this result was generalized to prime JBU -algebras in w19x. In the sequel, we shall use the following crucial result in the theory of topologically simple H U -algebras obtained in w5x. THEOREM 3.
E¨ ery topologically simple H U-algebra is centrally closed.
From the above theorem and w17, Lemma 4x, we deduce the following result. COROLLARY 1. E¨ ery dense subalgebra of a topologically simple H U-algebra is centrally closed. Let A be any nonassociative algebra. If a g A, then we denote by L a and R a the linear operators of left and right multiplication, respectively, by the element a in A. The multiplication algebra of A is defined as the subalgebra M Ž A. of LŽ A. Žthe algebra of all linear operators on A. generated by all left and right multiplication operators. It is easy to check that M Ž A. coincides with the subring of LŽ A. generated by all left and right multiplication operators. For a given subalgebra B of A, let us denote by MB Ž A. the subalgebra of M Ž A. generated by the multiplication operators L b and R b where b g B. Note that, for a normed algebra A, every element in M Ž A. is clearly a continuous linear operator on A.
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Furthermore, if B is a dense subalgebra of A, then the elements of M Ž B . are nothing but the restriction to B of those of MB Ž A.. This fact will be used in the sequel without previous comment. For a given continuous linear operator F on a Hilbert space we denote by F U its adjoint operator. Given an element a in an H U-algebra A we have LUa s L aU and RUa s R aU . Accordingly, for a dense subalgebra B of A, Ž MB Ž A..U s MBU Ž A. and M Ž A. becomes a self-adjoint subalgebra of the algebra of all continuous linear operator on the Hilbert space A. Let F be a ring homomorphism from an algebra A into an algebra B, that is an additive mapping from A into B satisfying F Ž ab. s F Ž a. F Ž b . for all a, b g A. For every a g A, we have F L a s LF Ž a. F and F R a s R F Ž a.F. Therefore the subring of those P in M Ž A. for which there exists Q in M Ž B . such that F P s QF, contains all left and right multiplication operators on A and therefore equals M Ž A.. Furthermore, if F is a ring homomorphism from A onto a dense subalgebra of a normed algebra B we obtain, for every P in M Ž A., a unique element f Ž P . in MF Ž A.Ž B . such that F P s f Ž P . F. So we can define a ring homomorphism F from M Ž A. onto MF Ž A.Ž B .. From the identity F P s f Ž P . F, it follows that if F is additionally injective, then f is a ring isomorphism from M Ž A. onto MF Ž A.Ž B .. In order to investigate the continuity properties of ring isomorphisms, we follow the traditional sliding hump procedure. To this end, we apply w25, Thm. 5x to build sequences having amazing properties, which allows us to put two powerful automatic continuity principles into action. THEOREM 4. Let B be a dense subalgebra of an infinite-dimensional topologically simple H U-algebra A. Then one of the following assertions holds: ŽC1. There exists sequences bn4 in BU and Pn4 in MB Ž A. such that PnU . . . P1U bn / 0 ŽC2.
U and Pnq1 PnU . . . P1U bn s 0
; n g N.
There exists a sequence Q n4 in MB Ž A. such that dim Q n Ž B . s 1
;n g N
Q n2 / 0
;n g N
Qn Qm s 0
if n - m.
Proof. BU is also a dense subalgebra of A, and Corollary 1 shows that it is centrally closed. Moreover, it is easy to check that BU is a topologically simple algebra. Finally, the result follows from w25, Thm. 5x, where the multiplication algebra was additionally generated by the identity operator, but this condition is unnecessary.
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3. AUTOMATIC CONTINUITY METHODS Note that, if F is an additive mapping between two Žreal or complex. vector spaces, then F is rational-linear and for rational-linear mappings between normed spaces continuity is equivalent to boundedness as for linear mappings. Moreover, we can measure the continuity of an additive mapping F between Banach spaces X and Y by considering the subset S Ž F . s y g Y : there exists x n ¬ 0 and F Ž x n . ¬ y 4 . It is easy to check that S Ž F . is a closed real-linear subspace of Y, which will be called the separating subspace for F. We observe that closed graph theorem remains true for additive mappings between Banach spaces and so the mapping F is continuous if, and only if, S Ž F . equals zero. Moreover, w21, Lemma 1.3x can be applied in order to obtain that, for any continuous linear operator G with domain Y but which may map into another Banach space Z, GF is continuous if, and only if, GŽ S Ž F .. s 0. Furthermore, if F is a ring homomorphism with dense range between normed algebras A and B, then S Ž F . is a real-linear ideal of B. For studying the continuity behaviour of a ring isomorphism acting between H U-algebras, we require two fundamental principles in the theory of automatic continuity stated in w23, 27x that remain true for additive mappings acting between Banach spaces. PROPOSITION 1 w23x. Let X be a Banach space, Sn4 a sequence of continuous linear operators from X into itself and R n4 be a sequence of continuous linear operators whose domain is a Banach space Y but which may map into others Banach spaces Yn . If F is a possibly discontinuous additi¨ e mapping from X into Y, such that R n FS1 . . . S m is continuous for m ) n, then R n FS1 . . . S n is continuous for sufficiently large n. PROPOSITION 2 w27x. Let X and Y be Banach spaces, S n4 a sequence of continuous linear operators from X into itself satisfying Sn Sm s 0 if n - m, and R n4 be a sequence of continuous linear operators from Y into itself. If F is a possibly discontinuous additi¨ e mapping from X into Y such that FS n y R n F is continuous for e¨ ery n g N, then FS n2 , and so R 2n F, is continuous for sufficiently large n.
4. RING ISOMORPHISMS OF TOPOLOGICALLY SIMPLE H U-ALGEBRAS Our main goal is to show that every ring isomorphism between infinitedimensional topologically simple H U-algebras is continuous. From this
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continuity, it will follows either the linearity or the conjugate-linearity of such isomorphism. LEMMA 1. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. If C1 is fulfilled for F Ž A., then F is continuous. Proof. For every n g N we put Pn s f Ž Sn . for a suitable Sn g M Ž A. and we define R nŽ b . s Ž b < bn . for all b g B. For every a g A and m, n g N we have R nFS1 . . . Sm a s Ž FS1 . . . Sm a < bn . s Ž f Ž S1 . . . Sm . F a < bn . s Ž f Ž S1 . . . . f Ž Sm . F a < bn . s Ž F a < PmU . . . P1U bn . , which equals zero if m ) n. From Proposition 1, it follows the continuity of the operator R nFS1 . . . S n , and hence the continuity of the functional a ¬ Ž F a < PnU . . . P1U bn . and A, for sufficiently large n. The linear subspace of B defined by J s b g B : a ¬ Ž F a < b . is continuous Ž and so real-linear . on A4 is closed by the Banach]Steinhaus theorem. It is easy to check that J is an ideal of B which contains the nonzero element PnU . . . P1U bn for sufficiently large n. Consequently J equals B by topological simplicity. The continuity of every functional a ¬ Ž F a < b . Ž b g B . shows that Ž S Ž F .< B . s 0 which implies that SŽ F . s 0 and hence the continuity of F. LEMMA 2. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. Then ker F s 0. Proof. Suppose the lemma were false. Then ker F q i ker F is a closed nonzero ideal in A and so ker F q i ker Fs A. Since Žker F q i ker F . A s Ž ker F . A q Ž ker F .Ž iA . ; ker F , it follow s th at A 2 s ker F q i ker FA ; ker F. Since A is the closure of the set Ý nks 1 a k bk : a k , bk g A, k s 1, . . . , n, n g N4 w8, Prop. 2Žviii.x, it may be concluded that ker Fs A. Let ker f denote the closure in M Ž A. of ker f . If a g A, then a s lim a n for a suitable sequence a n4 in ker F. Clearly L a s lim L a n , R a s lim R a n and further L a n , R a n g ker f ; n g N. Hence L a , R a g ker f. Since ker f is obviously a subalgebra of M Ž A. we conclude that ker fs M Ž A.. If F Ž A. fulfills ŽC1., then the preceding lemma shows that F is continuous and so ker F is closed. From what has already been proved, it follows that ker F s A and therefore F s 0, a contradiction.
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On the other hand, if F Ž A. satisfies ŽC2., then we have an element Q g MF Ž A.Ž B . with dim QŽ B . s 1 and Q 2 / 0. There are b1 , b 2 g B such that QŽ b . s Ž b < b1 . b 2 ; b g B. Furthermore Ž b 2 < b1 . / 0, since Q 2 / 0. Let S s Ž b 2 < b1 .y1 Q g MF Ž A.Ž B . and note that S / 0 and S 2 s S. There exists P g ker f such that f Ž P . s S and we take R g ker f such that 5 R y P 5 - 1. Consequently Ž IA y P q R . F s IA for a suitable invertible continuous linear operator F from A onto itself. Hence, we have F s F Ž IA y P q R . F s Ž F y F P q F R . F s Ž F y f Ž P . F q f Ž R . F . F s Ž F y SF . F s Ž IB y S . F F and so IB y S is a densely valued continuous linear operator from B onto itself. Since SŽ IB y S . s 0, we conclude that S s 0, a contradiction. THEOREM 5. Let A and B be infinite-dimensional topologically simple H U-algebras and F be a ring homomorphism from A onto a dense subalgebra of B. Then F is continuous and either a linear or a conjugate-linear isomorphism from A onto B. Proof. On account of Lemma 1, if F Ž A. fulfills ŽC1., then F is continuous. On the other hand, if F Ž A. fulfills ŽC2., then for each n g N we define Sn s fy1 Ž Q n .. Moreover, we set R n s Q n . According to Proposition 2, the operator Q n2 F is continuous and therefore S Ž F . ; kerŽ Q n2 ., for sufficiently large n. The ideal S Ž F . q i S Ž F . is contained in the closed linear subspace kerŽ Q n2 . for sufficiently large n. The topological simplicity of B implies that S Ž F . q i S Ž F . s 0 and so S Ž F . s 0, which shows the continuity of F. For studying the linearity of F, we follow the pattern established in w21, Thm. 7.6x. Let c 1 and c 2 be the operations of multiplication by the complex scalar i in A and B respectively. Then Ž Fy1c 2 F . 2 s c 12 s yIA and it is satisfied that Ž c 1Fy1c 2 F .Ž a. b s Ž Fy1c 2 Fc 1 .Ž a. b and bŽ c 1Fy1c 2 F .Ž a. s bŽ Fy1c 2 Fc 1 .Ž a. for all a, b g A. Thus Ž c 1Fy1c 2 F .Ž a. y Ž Fy1c 2 Fc 1 .Ž a. g AnnŽ A. s 0 for every a g A and so Fy1c 2 F and c 1 commute. If we define w s c 1Fy1c 2 F, then w is a continuous real-linear operator on A with w 2 s IA . Furthermore, w Ž ab. s w Ž a. b s a w Ž b . for all a, b g A. In particular, w satisfies w Ž iab. s iaw Ž b . s i w Ž ab. which shows that w is linear on A2 . Since w is continuous, w8, Prop. 2Žviii.x shows that w is a continuous linear operator on A. Let Aqs Ž w y IA . A and Ays Ž w q IA . A which are closed ideals of A, and A s Aq[ Ay. From the topological simplicity of A it follows that either A s Aq or A s Ay. Moreover, F is linear on Aq and conjugate-linear on Ay.
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Finally, to prove that F maps A onto B, we apply w7, Thm. 3.1x in the linear case. On the other hand, if F is a conjugate-linear operator, then we consider the topologically simple H U-algebra BX , the scalar product and inner product in BX being l ? a s l a and ² a, b : s Ž b < a. Ž l g C, a, b g B ., respectively, the sum, product, and involution being the same as those in B. F is a continuous linear homomorphism densely valued into BX . From w7, Thm. 3.1x, we deduce that F is onto, which completes the proof. EXAMPLES 2. Ž1. For any topologically simple associative H U -algebra A the symmetrized algebra Aq becomes a topologically simple Jordan H U -algebra Žsee w8x.. Every Jordan ring isomorphism F between two infinite-dimensional topologically simple associative H U-algebras A and B becomes a ring isomorphism from Aq onto Bq. From the above theorem we deduce that F is either a linear or a conjugate-linear continuous operator. Ž2. Any infinite-dimensional topologically simple associative H U -algebra A gives rise to a topologically simple Lie H U-algebra Ay, the multiplication in Ay being the usual Lie multiplication w x, y x s 12 Ž xy y yx ., the involution and inner product being the same as those in A Žsee w6x.. Every Lie ring isomorphism F between two infinite-dimensional topologically simple associative H U-algebras A and B becomes a ring isomorphism from Ay onto By and therefore is either a linear or a conjugatelinear continuous operator. Ž3. Let H1 and H2 be infinite-dimensional complex Hilbert spaces and J1 and J 2 conjugations on H1 and H2 , respectively, that is Jk is a conjugate-linear isometry from Hk onto itself satisfying Jk2 s IH k for k s 1, 2. Assume that t is an automorphism of the complex field and F any t-linear bijective map from H1 onto H2 for which the identity
Ž Fx < J2 Fy . s t Ž x < J1 y . holds for all x, y g H1. Then the Hilbert spaces A k s C [ Hk , with the products Ž l q x k .Ž m q y k . s lm q Ž x k < Jk y k . q l y k q m x k and involutions Ž l q x k .U s l q Jk Ž x k ., are infinite-dimensional topologically simple Jordan H U -algebras for k s 1, 2 Žsee w8x. and the map F Ž l, x . s Žt Ž l., Fx . becomes a ring isomorphism from A1 onto A 2 . On account of the above theorem, it follows that F is continuous and consequently F is continuous and t is either the identity or the conjugation on C. COROLLARY 2. Let A and B be infinite-dimensional topologically simple H U-algebras and F be any ring isomorphism from A onto B. Then F is either a linear or a conjugate-linear continuous isomorphism from A onto B and
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pro¨ ides either a linear or a conjugate-linear )-isomorphism C from A onto B. Furthermore there exists a positi¨ e constant r such that r C is isometric. Proof. From the preceding theorem, it follows that F is either a linear or a conjugate-linear continuous isomorphism from A onto B. If F is linear, then the remainder of the result follows from Theorem 2. If F is a conjugate-linear isomorphism we consider the topologically simple H U-algebra BX defined as at the end of the proof of Theorem 5. F is a linear isomorphism from A onto BX and consequently Theorem 2 assures the existence of a )-isomorphism C from A onto BX and a positive number r such that r ? C s r C is isometric. Therefore C is a conjugate-linear )-isomorphism from A onto B. Since the norm associated to the new inner product on B coincides with the original one, it follows that r C is isometric. Obviously in the finite-dimensional setting real-linearity and continuity are equivalent. Surprisingly, a discontinuous ring isomorphism between finite-dimensional topologically simple H U-algebras may be constructed. We give below an example of this situation. EXAMPLE 3. Let m be the automorphism from the subfield QŽ'2 . of C onto itself given by m Ž r q s'2 . s r y s'2 Ž r, s g Q.. Now we consider the set of all automorphisms of some subfield of C extending m , which is inductively ordered by extension. Let t : K ª K be a maximal element. We claim that t is an automorphism of the whole complex field. Otherwise there exists a complex number x with x f K. We apply w11, Thm. II.67x to obtain that K is algebraically closed and so x is transcendent over K. Therefore the automorphism t can be extended to an automorphism of KŽ x . and this contradiction proves our claim. Consequently t is a nonreal-linear Žand so discontinuous. automorphism of the complex field. Next, we show an intimate connection between Lebesgue measurability, continuity and linearity for additive maps. LEMMA 3. Let F be an additi¨ e mapping from R into itself. Then the following assertions are equi¨ alent: Ž1. Ž2. Ž3.
F is linear. F is continuous. F is measurable.
Proof. We only need to show that the third assertion implies the first one. Since R s D `ns 1 Fy1 Žwyn, n x., there exists n g N such that lŽ Fy1 Žwyn, n x.. ) 0, where l denotes the Lebesgue measure on R. By w22, Thm. 6.67x there exists d ) 0 such that x y d , d w; Fy1 Žwyn, n x. y
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Fy1 Žwyn, n x. ; Fy1 Žwy2 n, 2 n x., since F is additive. Consequently F Žx y d , d w. ; wy2 n, 2 n x and therefore F is bounded on a neighbourhood of zero, which shows that F is continuous on R. LEMMA 4. Let F be an additi¨ e mapping from a finite-dimensional Banach space X into a Banach space Y. Then the following assertions are equi¨ alent: Ž1. F is real-linear. Ž2. For e¨ ery element x in X and e¨ ery continuous linear functional f on Y, the mapping l ¬ R f Ž F Ž l x .. from R into itself is measurable. Proof. If Ž1. is fulfilled, then F is continuous and so the second assertion is obviously satisfied. According to the above lemma, if assertion Ž2. is satisfied, then R fF Ž l x . s R f Ž l Fx . for every x g X, every continuous linear functional f on Y, and every l g R. From the Hahn]Banach theorem, it follows that F is real-linear. PROPOSITION 3. Let A and B be topologically simple H U-algebras with A finite-dimensional and let F be a ring homomorphism from A into B. Then the following assertions are equi¨ alent: Ž1. F is real-linear and accordingly continuous. Ž2. There is a / 0 in A such that, for all b g B, the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable. Ž3. For all a g A and b g B the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable. Proof. On account of the preceding lemma, it suffices to prove that Ž3. follows from Ž2.. To this end we consider the set I of those a g A for which the map l ¬ R Ž F Ž l a.< b . from R into itself is measurable for every b g B. It is a simple matter to show that I is a nonzero ideal of A. Since A is finite-dimensional and topologically simple, it follows that I s A, which is the desired conclusion. Remark 1. It is known that any proof of the existence of nonmeasurable subsets of R requires, at some stage, uncountably many arbitrary choices of elements of R. The above connection between linearity and measurability shows that the use of Zorn lemma is unavoidable to construct discontinuous ring isomorphisms between finite-dimensional topologically simple H U-algebras. One can make the following somewhat vague statement. Any ring isomorphism whose values may be designated by elementary analytical constructions is measurable and consequently continuous. Loosely speaking: if you have a discontinuous ring isomorphism
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between topologically simple H U-algebras, then you cannot tell me which values it assumes, although you might be able to tell some of them. Although ring isomorphisms between finite-dimensional topologically simple H U-algebras may be non real-linear, next we prove that they satisfy a sort of linearity. THEOREM 6. Let F be a ring isomorphism between two centrally closed prime algebras A and B o¨ er a field K. Then there exists an automorphism t of K such that F Ž l a. s t Ž l. F a for all l g K and a g A. Proof. Let fix a nonzero element a in A. For every l g K, we claim that there exists taŽ l. g K such that F Ž l a. s taŽ l. F a. Assume that QŽ F a. s 0 with Q g M Ž B . and write P s fy1 Ž Q .. We have F Ž Pa. s QŽ F a. s 0 which shows that Pa s 0 and therefore 0 s F Ž l Pa. s F Ž P Ž l a.. s QŽ F Ž l a... Consequently, QŽ F a. ¬ QŽ F Ž l a.. is a well-defined linear mapping Žsay f . from the nonzero ideal I s M Ž B .Ž F a. into B satisfying f Ž by . s bf Ž y . and f Ž yb . s f Ž y . b for all y g I and b g B. Since B is centrally closed, there exists taŽ l. g K such that QŽ F Ž l a. y taŽ l. F a. s 0 for every Q g M Ž B . and so F Ž l a. s taŽ l. F a. Now we prove that the scalar taŽ l. is independent of which nonzero element a we choose to define it. Let b be a nonzero element in A. For all Q, QX g M Ž B ., we have Q Ž F Ž l b . . QX Ž F a . s F Ž fy1 Ž Q . Ž l b . . F Ž fy1 Ž QX . Ž a . . s F Ž fy1 Ž Q . Ž l b . fy1 Ž QX . Ž a . . s F Ž fy1 Ž Q . Ž b . fy1 Ž QX . Ž l a . . s F Ž fy1 Ž Q . Ž b . . F Ž fy1 Ž QX . Ž l a . . s Q Ž F b . QX Ž F Ž l a . . s Q Ž F b . QX Ž ta Ž l . F a . s Q Ž ta Ž l . F b . QX Ž F a . . The ideals M Ž B .ŽtaŽ l. F b y F Ž l b .. and M Ž B .Ž F a. have zero product. By primeness M Ž B .ŽtaŽ l. F b y F Ž l b .. equals zero, since M Ž B .Ž F a. / 0. Thus taŽ l. F b y F Ž l b . s 0 and therefore taŽ l. s t b Ž l.. We will denote this scalar by tF Ž l.. Now it is easy to check that tF Ž l q m . s tF Ž l. q tF Ž m . and tF Ž lm . s tF Ž l.tF Ž m . for all l, m g K and tF y1 tF s tFtF y 1 s IK , which shows that tF is an automorphism of the field K. Remark 2. It is easy to check that if in the preceding theorem A and B are in addition normed algebras and F is continuous, then so is t .
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Remark 3. From the preceding theorem, we deduce that ring isomorphisms between centrally closed prime algebras preserve the algebraic dimension. EXAMPLE 4. Every ring isomorphism between any of the algebras showed in Examples 1 is automatically t-linear for a suitable automorphism t of C. Since topologically simple H U-algebras are centrally closed prime algebras, we get the following. COROLLARY 3. Let F be a ring isomorphism between two topologically simple H U-algebras A and B. Then F is t-linear for a suitable automorphism t of the complex field. EXAMPLE 5. Schue proved in w20x that finite-dimensional simple Lie complex algebra can be structured as H U-algebras. On account of the preceding corollary, we deduce that every ring isomorphism between such algebras is automatically t-linear for a suitable automorphism t of C. COROLLARY 4. Ring isomorphic topologically simple associati¨ e H U-algebras are automatically )-isomorphic. Proof. Let A and B topologically simple associative H U-algebras and F be a ring isomorphism from A onto B. By the preceding result, F is t-linear for a suitable automorphism t of the complex field. By w16, Thm. 4.10.32x, A is )-isomorphic to a ‘‘full matrix algebra’’ M G . The mapping Ž lgn .g , n g G ¬ Žty1 Ž lgn ..g , n g G is a ty1 -linear automorphism of M G Žnote that, if G is infinite, then t is either the identity or the conjugation on the complex field. which gives a ty1 -linear automorphism C of A. FC is a linear isomorphism from A onto B which provides, by Theorem 2, a )-isomorphism from A onto B. Note that every topologically simple associative H U-algebra is prime and, if it has dimension greater than one, then it contains nontrivial idempotents. Consequently, w14, Corollaryx now yields the following. LEMMA 5. E¨ ery multiplicati¨ e bijecti¨ e mapping from a topologically simple associati¨ e H U-algebra with dimension greater than one onto an arbitrary associati¨ e ring is automatically additi¨ e. EXAMPLE 6. It should be noted that the conclusion of the above-established result fails if the H U-algebra is one-dimensional. Indeed, the inversion mapping l ¬ ly1 Žwith 0 ¬ 0. from C into itself is multiplicative and nonadditive. In spite of the preceding example, we get the following multiplicative characterization of the topologically simple associative H U-algebras.
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THEOREM 7. Multiplicati¨ ely isomorphic topologically simple associati¨ e H U-algebras are automatically )-isomorphic. Proof. Let A and B topologically simple associative H U-algebras and F a multiplicative bijective mapping from A onto B. If A and B have dimension one, then both of them are )-isomorphic to the complex field and consequently the conclusion in the theorem is true. If either A or B has dimension greater than one, then we conclude from Lemma 5 that F is a ring isomorphism, hence that A and B are )-isomorphic by Corollary 4. Remark 4. It is worth pointing out that multiplicative bijective mapping between topologically simple associative H U-algebras preserves the algebraic dimension. Questions 1. The above established results lead naturally to the following questions: Ž1. Are ring isomorphic topologically simple Žnonassociative. H U -algebras automatically )-isomorphic? Ž2. Are multiplicatively isomorphic topologically simple Žnonassociative. H U -algebras automatically )-isomorphic?
5. RING ISOMORPHISMS OF H U-ALGEBRAS WITH ZERO ANNIHILATOR The study of ring isomorphisms between H U-algebras with zero annihilator can be reduced to the case of ring isomorphisms between topologically simple H U-algebras. LEMMA 6. Let A and B be nonassociati¨ e rings and F be a multiplicati¨ e mapping from A onto B. Then F Ž0. s 0. Proof. Put 0 s F Ž a. for a suitable a g A. We have F Ž0. s F Ž a0. s F Ž a. F Ž0. s 0F Ž0. s 0. LEMMA 7. Let A and B be H U-algebras with zero annihilator and F be a multiplicati¨ e bijecti¨ e mapping from A onto B. If I is a closed ideal in A, then F Ž I . is an ideal of B. Proof. We only need to show that F Ž I . q F Ž I . ; F Ž I . and lF Ž I . ; F Ž I . for every l g C.
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Since F is surjective there exists a subset J of A such that F Ž J . s F Ž I . q F Ž I .. On account of the preceding lemma, we have F Ž JI H . s F Ž J . F Ž I H . s Ž F Ž I . q F Ž I . . F Ž I H . ; F Ž II H . q F Ž II H . s 0 and hence that JI H s 0. From w8, Prop. 2Žvii.x, it follows that J ; I and therefore F Ž I . q F Ž I . ; F Ž I .. The set lF Ž I . may be handled in much the same way. Indeed, let J ; A satisfying F Ž J . s lF Ž I .. Since F Ž JI H . s F Ž J . F Ž I H . s lF Ž I . F Ž I H . s lF Ž II H . s 0, we have JI H s 0 and consequently J ; I. Hence lF Ž I . ; F Ž I .. LEMMA 8. Let A and B be H U-algebras with zero annihilator and F be a multiplicati¨ e bijecti¨ e mapping from A onto B. If I is a minimal closed ideal in A, then F Ž I . is a minimal closed ideal of B. Proof. If F Ž I . J s 0 for every minimal closed ideal J of B, then by Theorem 1 it follows that F Ž I . B s 0. Therefore F Ž I . ; AnnŽ B . s 0 and so I s 0 which leads to a contradiction. Consequently there exists a minimal closed ideal J of B for which F Ž I . J / 0, equivalently IFy1 Ž J . / 0. We claim that F Ž I . ; J. Since J is a topologically simple H U -algebra Žsee Theorem 1. and F Ž I . l J is a nonzero closed ideal, we conclude that F Ž I . l J s J and therefore J ; F Ž I . . Decompose F Ž I . into an orthogonal sum J [ J X with J X ; J H . Since Fy1 Ž J . Fy1 Ž J X . s Fy1 Ž JJ X . s 0, we have
Ž Fy1 Ž J . l I .Ž Fy1 Ž J X . l I . ; Fy1 Ž J . Fy1 Ž J X . ; Fy1 Ž J . Fy1 Ž J X . s 0, and the primeness of I shows that Fy1 Ž J X . l I s 0 Žnote that 0 / IFy1 Ž J . ; Fy1 Ž J . l I .. Consequently, 0 s F Ž Fy1 Ž J X . I . s J X F Ž I . and J XF Ž I . s 0. Hence 0 s J XF Ž I . s J X Ž J q J X . s J X J X , and the semiprimeness of B shows that J X s 0. Thus F Ž I . s J, and the claim follows. The same proof works for the nonzero closed ideal Fy1 Ž J . l I of I, which gives Fy1 Ž J . ; I. Consequently, F Ž I . s J as required. The above-established result shows that every ring isomorphism between two given H U-algebras with zero annihilator provide ring isomorphisms
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between topologically simple H U-algebras. The preceding section works in this situation, and therefore we can formulate the following result. THEOREM 8. Let A and B be H U-algebras with zero annihilator and F be a ring isomorphism from A onto B. Then A and B decompose into orthogonal sums of closed ideals A s Aq[ Ay[ A1 [ ??? [ A n and B s Bq[ By[ B1 [ ??? [ Bn , with A1 , . . . , A n and B1 , . . . , Bn finite-dimensional, in such a way that the restriction FAq is a continuous linear isomorphism from Aq onto Bq, the restriction FAy is a continuous conjugate-linear isomorphism from Ay onto By, and, for k s 1, . . . , n, the restriction FA k is a t k-linear isomorphism from A k onto Bk for a suitable automorphism t k of the complex field. Proof. Let I be a minimal closed ideal of A. The restriction FI is a ring isomorphism from I onto the minimal closed ideal F Ž I . of B. On account of Theorem 1, I and F Ž I . are topologically simple H U -algebras. If in addition I has infinite dimension, then Corollary 2 shows that F is either a linear or a conjugate-linear continuous isomorphism from I onto F Ž I .. We claim that the set of those finite-dimensional minimal closed ideals on which F is discontinuous is finite. Suppose the claim were false. Then there exists a sequence In4 of pairwise different finite-dimensional minimal closed ideals in A on each of which F is discontinuous. For every n g N, MI nŽ A. acts topologically irreducibly on In and consequently it acts irreducibly on In , since In is finite-dimensional. Therefore, for every n g N, there exists Sn g MI nŽ A. with S n2 / 0. Since In Im s 0 if n / m, it follows that Sn Sm s 0 if n / m. Proposition 2 now shows that the operator f Ž Sn2 . FI n is continuous, and so S Ž FI n . ; ker f Ž Sn2 ., for sufficiently large n. Therefore the ideal S Ž FI n . q i S Ž FI n . is contained in the closed linear subspace ker f Ž Sn2 ., for sufficiently large n. Since f Ž S n2 . / 0 and F Ž In . is topologically simple, we conclude that S Ž FI n . s 0 and hence that FI n is continuous for sufficiently large n. This contradicts our assumption. Let us denote by Iq the set of all minimal closed ideals of A on which F is a continuous linear map, by Iy the set of those on which F is a continuous conjugate-linear map, and by A1 , . . . , A n the finitely many exceptional minimal closed ideals. Furthermore, we write Jqs F Ž I . : I g Iq 4 , Jys F Ž I . : I g Iy 4 , and B1 s F Ž A1 ., . . . , Bn s F Ž Bn .. Finally, we define As and Bs as the closure of the orthogonal sum of the ideals in Is and Js , respectively, for s g q, y4 . let A C s Aq[ Ay and BC s Bq[ By. Since F Ž A C . Bk s F Ž A C . F Ž A k . s F Ž A C A k . s 0 for k s 1, . . . , n, w8, Prop. 2Žvii.x shows that F Ž A C . ; F nks1 BkH s BC . Consequently S Ž FA C . ; BC . Let b g S Ž FA C .. Then b s lim F a n for a suitable sequence
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a n4 in A C converging to zero. For every minimal closed ideal I of A on which F is continuous and every a g I, we have bF a s lim F a nF a s lim F Ž a n a . s 0. Hence S Ž FA C . F Ž I . s 0. From this we deduce that S Ž FA C . BC s 0 and w8, Prop. 2Žvii.x now leads to S Ž FA C . ; BCH . From what has already been proved, it may be concluded that S Ž FA C . s 0, which shows that F is continuous on A C . From this we deduce that F Ž Aq . ; Bq and F Ž Ay . ; By. We now apply the preceding argument again, with F replaced by Fy1 , to obtain that Fy1 Ž Bq . ; Aq and Fy1 Ž By . ; Ay, which completes the proof. COROLLARY 5. Ring isomorphic associati¨ e H U-algebras with zero annihilator are automatically )-isomorphic. Proof. Let F be a ring isomorphism between two given associative H U-algebras A and B with zero annihilator, and consider the decomposition given in the preceding theorem. By Theorem 2, the H U-algebras Aq and Bq are )-isomorphic. By Corollary 4, every H U-algebra A k is )-isomorphic to Bk for k s 1, . . . , n. It is a simple matter to construct a conjugate-linear automorphism C of Ay Žtake into account the proof of Corollary 4.. FAy C is a linear isomorphism from Ay onto By. Consequently Ay and By are )-isomorphic, which completes the proof. LEMMA 9. Let A be an associati¨ e H U-algebra with zero annihilator which has no one-dimensional minimal closed ideals and let F be a multiplicati¨ e bijecti¨ e mapping from A onto an arbitrary associati¨ e ring B. Then F is automatically additi¨ e. Proof. Let us denote by I the set of all minimal closed ideals of A. Every I g I is a topologically simple associative H U-algebra with dimension greater than one and therefore it contains a nontrivial idempotent e I . We shall have established the lemma if we prove that A satisfies the conditions in the Martindale Theorem w14x for the set e I : I g I 4 of idempotents. If e I Aa s 0 for each I g I , then Ž Ae I A. a s 0 and consequently Ae I Aa s 0 for each I g I . Since Ae I As I, we conclude that Ia s 0 for each I g I and therefore Aa s 0 ŽTheorem 1.. Since A has zero annihilator we conclude that a s 0. If e I ae I AŽ1 y e I . s 0 for a given I g I , then Ž e i ae I . I Ž1 y e I . I s 0 and so Ž e I ae I .I Ž 1 y e I . Is 0. Since I Ž 1 y e I . Is I, we have Ž e I aeI . I s 0, and hence e I aeI g AnnŽ I . s 0. THEOREM 9. Let A and B be H U-algebras with zero annihilator and finitely many one-dimensional minimal closed ideals. If there exists a multiplicati¨ e bijecti¨ e mapping F from A onto B, then A and B are automatically )-isomorphic.
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Proof. Let us denote by I1 and J1 the set of all one-dimensional minimal closed ideals of A and B, respectively, and let I2 and J2 denote the sets of those which have dimension greater than one. According to Lemma 8 and Remark 4, we have
F Ž I . : I g I1 4 s J : J g J1 4
and
F Ž I . : I g I2 4 s J : J g J2 4 .
We write A k and Bk Ž k s 1, 2. for the closure in A and B, respectively, of the orthogonal sum of the elements of Ik and Jk , respectively. For every ideal J g J1 , we put J s F Ž I . for a suitable I g I1 and therefore F Ž A 2 . J s F Ž A 2 . F Ž I . s F Ž A 2 I . s 0. Hence F Ž A 2 . B1 s 0 and w8, Prop. 2Žvii.x shows that F Ž A 2 . ; B1H s B2 . The same reasoning applies to Fy1 gives Fy1 Ž B2 . ; A 2 and consequently F Ž A 2 . s B2 . According to the preceding lemma, F is a ring isomorphism from A 2 onto B2 and Corollary 5 shows that A 2 and B2 are )-isomorphic. Since A1 and B1 are obviously )-isomorphic, it may be concluded that A and B are )-isomorphic. Question 2. Are multiplicatively isomorphic associative H U-algebras with zero annihilator and infinitely many one-dimensional minimal closed ideals )-isomorphic?
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