ISRN AlgebraVolume 2012 (2012), Article ID 328752, 11 pagesdoi:10.5402/2012/328752

Research Article

On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

Mohamed Benslimane and Abdelhadi Moutassim

Département de Mathématiques et Informatique, Faculté des Sciences, B.P. 2121, Tétouan, Morocco

Received 17 April 2012; Accepted 30 May 2012

Academic Editors: A. Jaballah, A. Kiliçman, D. Sage, K. P. Shum, F. Uhlig, A. Vourdas, and H. You

Copyright © 2012 Mohamed Benslimane and Abdelhadi Moutassim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

1. Introduction

Let A be a real or complex algebra not necessarily associative or finite dimensional. Assuming that a vector space A is endowed with a pre-Hilbert norm satisfying for all . Zalar (1995, [1]) proved that. (1)If is a real alternative algebra containing a unit element such that , then is finite dimensional and is isomorphic to , or . (2)If is a real associative algebra satisfying , then is finite dimensional and is isomorphic to , or . (3)If is a complex normed algebra containing a unit element such that , then is finite dimensional and is isomorphic to . These results were extended, respectively, to the following cases. (1)If is a real alternative algebra containing a nonzero central element such that , then is finite dimensional and is isomorphic to , or (2008, [2]). (2)If is a real alternative algebra satisfying , then is finite dimensional and is isomorphic to , or (2008, [2]). (3)If is a complex normed algebra without divisors of zero and containing an invertible element such that , then is finite dimensional and is isomorphic to (2010, [3]). In the present paper, we extend the above results to more general situation. Indeed, we prove that, if is a real or complex pre-Hilbert algebra satisfying for all , then is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero and satisfying for all (Theorem 3.5). (2) is a real weakly alternative algebra without divisors of zero and containing a nonzero central element such that for all (Theorem 3.7). (3) is a complex powers associative algebra satisfying for all (Theorem 4.8). In the first two cases is isomorphic to or and is isomorphic to in the last two cases. This last allows us to show that if is a complex pre-Hilbert noncommutative Jordan algebra (resp., flexible algebraic algebra or Jordan algebra) satisfying for all , then is finite dimensional and is isomorphic to (Theorems 4.9 and 4.10). Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra (weakly alternative algebra) with divisors of zero and satisfying for all .

2. Notation and Preliminary Results

Throughout the paper, the word algebra refers to a nonnecessarily associative algebra over or .

Definitions 1. Let be an arbitrary algebra and is a field of characteristic not .(1)(i) is called alternative if it is satisfied the identities and (where means associator), for all (1966, [4]).(ii) is called a powers associative if, for every in , the subalgebra generated by is associative.(iii) is called flexible if for all .(iv) is called a Jordan algebra if it is commutative and satisfied the Jordan identity: (J) for all .(v) is called a noncommutative Jordan algebra if it is flexible and satisfied the Jordan identity (J).(vi) is called weakly alternative if it is a noncommutative Jordan algebra and satisfied the identity (where means commutator). An alternative algebra or Jordan algebra is evidently weakly alternative.(vii) is called quadratic if it has an identity element and satisfied the identity for all and .(2)(viii) We say that is algebraic if, for every in , the subalgebra of generated by is finite dimensional (1947, [5]).(ix) A symmetric bilinear form over is called a trace form if for all .(x) is termed normed (resp., absolute valued) if it is endowed with a space norm such that (resp., ), for all .(xi) is called a pre-Hilbert algebra if it is endowed with a space norm comes from an inner product .(xii) We mean by a nonzero central element in , a nonzero element which commute with all elements of the algebra .The most natural examples of absolute valued algebras are (the algebra of Hamilton quaternion) and (the algebra of Cayley numbers), with norms equal to their usual absolute values (1991, [6]) and (2004, [7]). The algebra (1949, [8]) was obtained by replacing the product of with the one defined by , where means the standard involution of .

We have the following very known results.

Lemma 2.1 (see [4]). Let be a powers associative algebra over and without divisors of zero. If is a nonzero idempotent in , then has an identity element .

Proposition 2.2 (see [9]). If is a set of commuting elements in a flexible algebra over , then the subalgebra generated by the is commutative.

Proposition 2.3 (see [10]). Let be a noncommutative Jordan algebra over , then is a powers associative algebra.

Lemma 2.4 (see [11]). Let be a quadratic algebra over . Then flexible if and only if is symmetric and the following equivalent statements hold. (1) is a trace form over .(2) is a trace over .(3) = 0 for every .

Theorem 2.5 (see [4]). The subalgebra generated by any two elements of an alternative algebra is associative.

We need the following results.

Theorem 2.6 (see [1]). Let be a real pre-Hilbert associative algebra satisfying for all . Then is finite dimensional and is isomorphic to , or .

Theorem 2.7 (see [2]). Let be a real pre-Hilbert commutative algebra without divisors of zero and satisfying for all . Suppose that containing a nonzero central element such that for all . Then is isomorphic to , or .

Theorem 2.8 (see [1]). Let be a real pre-Hilbert alternative algebra with identity . Suppose that for all and . Then is isomorphic to , or .

3. Real Pre-Hilbert Weakly Alternative Algebras

In this subparagraph, we prove that, if is a real pre-Hilbert algebra satisfying for all . Then is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a real Jordan algebra without divisors of zero. In the first case is isomorphic to , or , and is isomorphic to or in the last case. Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

Lemma 3.1 (see [12]). Let be a real pre-Hilbert algebra with identity such that for all and let then. (1). (2) for all .

Remark 3.2. (i) The product , for all , provides the structure of an anticommutative algebra.(ii) If is flexible, then for all .

Proof. (i) Let , we have (ii) As is a flexible algebra, then This implies that for all , and by Lemma 3.1, we have . Thus, .

Theorem 3.3. Let be a real pre-Hilbert weakly alternative algebra with identity and without divisors of zero. Suppose that for all and . Then is finite dimensional and is isomorphic to , or .

Proof. It is sufficient to prove that is an alternative algebra. Let such that , according to Lemma 3.1 we have This implies that So As has nonzero divisors, then Therefore, . Now we take two arbitrary elements , and let . Or , then Let and two elements in , with and , we have . Therefore , thus . So is a left alternative algebra. Now we show that is a right alternative algebra, if are two orthogonal elements. Then And (Remark 3.2), thus, Similarly, we prove that for all , then is a right alternative algebra. Thus, is an alternative algebra, the result ensues then of Theorem 2.8.

Corollary 3.4. Let be a real pre-Hilbert Jordan algebra with identity and without divisors of zero. Suppose that for all and , then is finite dimensional and is isomorphic to or .

Theorem 3.5. Let be a real pre-Hilbert weakly alternative algebra without divisors of zero. Suppose that for all , then is finite dimensional and is isomorphic to , or .

Proof. is a powers associative algebra (Proposition 2.3) then the subalgebra of , generated by , is associative and verifying the conditions of Theorem 2.6. Therefore, is isomorphic to or , thus there is a nonzero idempotent such that ; that is, is a unital algebra of unit (Lemma 2.1). So the result is a consequence of Theorem 3.3.

Corollary 3.6. Let be a real pre-Hilbert Jordan algebra without divisors of zero. Suppose that for all , then is finite dimensional and is isomorphic to or .

We give an extension of Theorem 3.3.

Theorem 3.7. Let be a real pre-Hilbert weakly alternative algebra without divisors of zero and satisfying for all . Suppose that containing a nonzero central element such that for all . Then is finite dimensional and is isomorphic to , or .

Proof . Let , the subalgebra of generated by is commutative. Theorem 2.7 implies that , thus the result is a consequence of Theorem 3.5.

Corollary 3.8. Let be a real pre-Hilbert Jordan algebra without divisors of zero and satisfying for all . Suppose that contains a nonzero central element such that for all . Then is finite dimensional and is isomorphic to or .

Remark 3.9. In the previous results the hypothesis without divisors of zero is necessary. The following example proves it.Let be an infinite-dimensional real Hilbert space, we define the multiplication on the vector space by: And the scalar product by So is a commutative algebra satisfying and for all . Indeed, we put and . We have Then , moreover, Then Thus, Similarly, Thus, From the two equalities (3.15) and (3.17), we conclude that ; that is, for all . This implies that is an infinite-dimensional real pre-Hilbert Jordan (weakly alternative) algebra with identity satisfying and has a zero divisors. Indeed, let and be two orthogonal nonzero elements in , as defined multiplication of , we have . Hence, is an algebra with zero divisors.

4. Complex Pre-Hilbert Noncommutative Jordan Algebras Satisfying

We show that if is a noncommutative Jordan complex pre-Hilbert algebra satisfying for all , then is finite dimensional and is isomorphic to .

4.1. Complex Pre-Hilbert Alternative Algebras Satisfying

We need the following results.

Proposition 4.1 (see [3]). Let be a complex pre-Hilbert commutative associative algebra satisfying for all . Then is finite dimensional and is isomorphic to .

Theorem 4.2 (see [3]). Let be a complex pre-Hilbert algebra with identity . Suppose that for all . Then is finite dimensional and is isomorphic to .

Lemma 4.3 (see [3]). Let be a complex pre-Hilbert commutative algebra satisfying for all . Then has nonzero divisors.

Theorem 4.4 (see [3]). Let be a complex pre-Hilbert commutative algebraic algebra satisfying for all . Then is finite dimensional and is isomorphic to .

Lemma 4.5. Let be a complex pre-Hilbert alternative algebra satisfying for all . Then has nonzero divisors.

Proof. Let be a nonzero element in and let an element in such that . The subalgebra of generated by is associative (Theorem 2.5). We have then . Thus, is a commutative and associative, therefore, the Proposition 4.1 complete the demonstration.

Theorem 4.6. Let be a complex pre-Hilbert alternative algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. Let , the subalgebra of generated by is commutative and associative (Theorem 2.5). Proposition 4.1 proves that is isomorphic to , then there exists a nonzero idempotent . According to Theorem 4.2 it is sufficient to prove that is a unit element of . Let , we have and . As is without divisors of zero (Lemma 4.5), then . Thus, is finite dimensional and is isomorphic to .

4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying

In this subparagraph we show that if (, ) is a complex pre-Hilbert powers associative algebra (resp., flexible algebraic algebra, noncommutative Jordan algebra, or weakly alternative algebra) satisfying for all . Then is finite dimensional and is isomorphic to .

We have the following importing result.

Lemma 4.7. Let be a complex pre-Hilbert powers associative algebra satisfying for all . Then has nonzero divisors.

Proof. Let be a nonzero element in , the subalgebra of is associative. According to Theorem 4.6, is isomorphic to . Therefore, there exist a nonzero idempotent and such that . Suppose there is a nonzero element , as is isomorphic to (Theorem 4.6), then there exist a nonzero idempotent and such that . We have , and This implies that , because . Thus, or This is absurd and hence, has nonzero divisors.

Theorem 4.8. Let be a complex pre-Hilbert powers associative algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. According to Lemma 4.7, has a nonzero divisors. Let be a nonzero element in , then the subalgebra of is associative. Theorem 4.6 implies that is isomorphic to . Hence, containing a nonzero idempotent, this gives that has a unit element (Lemma 2.1). The result is a consequence of Theorem 4.2.

Theorem 4.9. Let be a complex pre-Hilbert flexible algebraic algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. Let be a nonzero element, according to Proposition 2.2 and Lemma 4.3, the subalgebra of is commutative, algebraic, and without divisors of zero. Thus is isomorphic to (Theorem 4.4). This implies that is a powers associative algebra, then the result is a consequence of Theorem 4.8.

We state the main theorem.

Theorem 4.10. Let be a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. Proposition 2.3 implies that is a powers associative algebra, and hence, is isomorphic to (Theorem 4.8).

Corollary 4.11. Let be a complex pre-Hilbert weakly alternative algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. is a noncommutative Jordan algebra. By Theorem 4.10, is finite dimensional and is isomorphic to .

Corollary 4.12. Let be a complex pre-Hilbert Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to .

Proof. is a weakly alternative algebra. By Corollary 4.11, is finite dimensional and is isomorphic to .

Acknowledgment

The authors are very grateful to professor A. M. Kaidi for his advice and help. This paper is dedicated to the memory of professor Khalid Bouhya.

References

B. Zalar, “On Hilbert spaces with unital multiplication,” Proceedings of the American Mathematical Society, vol. 123, no. 5, pp. 1497–1501, 1995.

A. Moutassim and A. Rochdi, “Sur les algèbres préhilbertiennes vérifiant ,” Advances in Applied Clifford Algebras, vol. 18, no. 2, pp. 269–278, 2008.

M. R. Hilali, A. Moutassim, and A. Rochdi, “-algebres normées préhilbertiennes vérifiant ,” Advances in Applied Clifford Algebras, vol. 20, no. 1, pp. 33–41, 2010.

R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, NY, USA, 1966.

A. A. Albert, “Absolute valued real algebras,” Annals of Mathematics. Second Series, vol. 48, pp. 495–501, 1947.

H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch et al., Numbers, vol. 123 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1991.

A. Rodriguez, “Absolute-valued algebras, and absolute-valuable Banach spaces,” in Advanced Courses of Mathematical Analysis I, pp. 99–155, World Scientific, Hackensack, NJ, USA, 2004.

A. A. Albert, “A note of correction,” Bulletin of the American Mathematical Society, vol. 55, p. 1191, 1949.

G. M. Benkart, D. J. Britten, and J. M. Osborn, “Real flexible division algebras,” Canadian Journal of Mathematics, vol. 34, no. 3, pp. 550–588, 1982.

H. Braun and M. Koecher, Jordan-Algebren, Springer, Berlin, Germany, 1966.

J. M. Osborn, “Quadratic division algebras,” Transactions of the American Mathematical Society, vol. 105, pp. 202–221, 1962.

A. Moutassim, “Quelques résultats sur les algèbres flexibles préhilbertiennes sans diviseurs de zéro vérifiant ,” Advances in Applied Clifford Algebras, vol. 18, no. 2, pp. 255–267, 2008.