About this Journal Submit a Manuscript Table of Contents 
ISRN Algebra
VolumeΒ 2012Β (2012), Article IDΒ 328752, 11 pages
doi:10.5402/2012/328752
Research Article

On Pre-Hilbert Noncommutative Jordan Algebras Satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2

Mohamed BenslimaneΒ 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.Β Jaballah, A.Β Kiliçman, D.Β Sage, K. P.Β Shum, F.Β Uhlig, A.Β Vourdas, and H.Β You

Copyright Β© 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 𝐴 be a real or complex algebra. Assuming that a vector space 𝐴 is endowed with a pre-Hilbert norm β€– β‹… β€– satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . We prove that 𝐴 is finite dimensional in the following cases. (1) 𝐴 is a real weakly alternative algebra without divisors of zero. (2) 𝐴 is a complex powers associative algebra. (3) 𝐴 is a complex flexible algebraic algebra. (4) 𝐴 is a complex Jordan algebra. In the first case 𝐴 is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 , and 𝐴 is isomorphic to β„‚ in the last three cases. These last cases permit us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 .

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 β€– β‹… β€– satisfying β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Zalar (1995, [1]) proved that. (1)If 𝐴 is a real alternative algebra containing a unit element 𝑒 such that β€– 𝑒 β€– = 1 , then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 . (2)If 𝐴 is a real associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 , then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , or ℍ . (3)If 𝐴 is a complex normed algebra containing a unit element 𝑒 such that β€– 𝑒 β€– = 1 , then 𝐴 is finite dimensional and is isomorphic to β„‚ . These results were extended, respectively, to the following cases. (1)If 𝐴 is a real alternative algebra containing a nonzero central element π‘Ž such that β€– π‘Ž π‘₯ β€– = β€– π‘Ž β€– β€– π‘₯ β€– , then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 (2008, [2]). (2)If 𝐴 is a real alternative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 , then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , or ℍ (2008, [2]). (3)If 𝐴 is a complex normed algebra without divisors of zero and containing an invertible element 𝑣 such that β€– 𝑣 π‘₯ β€– = β€– π‘₯ 𝑣 β€– = β€– 𝑣 β€– β€– π‘₯ β€– , then 𝐴 is finite dimensional and is isomorphic to β„‚ (2010, [3]). In the present paper, we extend the above results to more general situation. Indeed, we prove that, if 𝐴 is a real or complex pre-Hilbert algebra satisfying β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional in the following cases. (1) 𝐴 is a real weakly alternative algebra without divisors of zero and satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 (Theorem 3.5). (2) 𝐴 is a real weakly alternative algebra without divisors of zero and containing a nonzero central element π‘Ž such that β€– π‘Ž π‘₯ β€– = β€– π‘Ž β€– β€– π‘₯ β€– for all π‘₯ ∈ 𝐴 (Theorem 3.7). (3) 𝐴 is a complex powers associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 (Theorem 4.8). In the first two cases 𝐴 is isomorphic to ℝ , β„‚ , ℍ or 𝕆 and 𝐴 is isomorphic to β„‚ in the last two cases. This last allows us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra (resp., flexible algebraic algebra or Jordan algebra) satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ (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 β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 .

2. Notation and Preliminary Results

Throughout the paper, the word algebra refers to a nonnecessarily associative algebra over ℝ or β„‚ .

Definitions 1. Let 𝐡 be an arbitrary algebra and 𝐾 is a field of characteristic not 2 .(1)(i) 𝐡 is called alternative if it is satisfied the identities ( 𝑦 , π‘₯ , π‘₯ ) = 0 and ( π‘₯ , π‘₯ , 𝑦 ) = 0 (where ( β‹… , β‹… , β‹… ) means associator), for all π‘₯ , 𝑦 ∈ 𝐡 (1966, [4]).(ii) 𝐡 is called a powers associative if, for every π‘₯ in 𝐡 , the subalgebra 𝐡 ( π‘₯ ) generated by π‘₯ is associative.(iii) 𝐡 is called flexible if ( π‘₯ , 𝑦 , π‘₯ ) = 0 for all π‘₯ , 𝑦 ∈ 𝐡 .(iv) 𝐡 is called a Jordan algebra if it is commutative and satisfied the Jordan identity: (J) ( π‘₯ 2 , 𝑦 , π‘₯ ) = 0 for all π‘₯ , 𝑦 ∈ 𝐡 .(v) 𝐡 is called a noncommutative Jordan algebra if it is flexible and satisfied the Jordan identity (J).(vi) 𝐡 is called weakly alternative if it is a noncommutative Jordan algebra and satisfied the identity ( π‘₯ , π‘₯ , [ π‘₯ , 𝑦 ] ) = 0 (where [ β‹… , β‹… ] means commutator). An alternative algebra or Jordan algebra is evidently weakly alternative.(vii) 𝐡 is called quadratic if it has an identity element 𝑒 and satisfied the identity π‘₯ 2 = 𝛼 𝑒 + 𝛽 π‘₯ for all π‘₯ ∈ 𝐡 and 𝛼 , 𝛽 ∈ 𝕂 .(2)(viii) We say that 𝐡 is algebraic if, for every π‘₯ in 𝐡 , the subalgebra 𝐡 ( π‘₯ ) of 𝐡 generated by π‘₯ is finite dimensional (1947, [5]).(ix) A symmetric bilinear form ( β‹… , β‹… ) over 𝐡 is called a trace form if ( π‘₯ 𝑦 , 𝑧 ) = ( π‘₯ , 𝑦 𝑧 ) for all π‘₯ , 𝑦 , 𝑧 ∈ 𝐡 .(x) 𝐡 is termed normed (resp., absolute valued) if it is endowed with a space norm β€– β‹… β€– such that β€– π‘₯ 𝑦 β€– ≀ β€– π‘₯ β€– β€– 𝑦 β€– (resp., β€– π‘₯ 𝑦 β€– = β€– π‘₯ β€– β€– 𝑦 β€– ), for all π‘₯ , 𝑦 ∈ 𝐡 .(xi) 𝐡 is called a pre-Hilbert algebra if it is endowed with a space norm comes from an inner product ( β‹… ∣ β‹… ) .(xii) We mean by a nonzero central element in 𝐡 , a nonzero element which commute with all elements of the algebra 𝐡 .
The most natural examples of absolute valued algebras are ℝ , β„‚ , βˆ— β„‚ , ℍ (the algebra of Hamilton quaternion) and 𝕆 (the algebra of Cayley numbers), with norms equal to their usual absolute values (1991, [6]) and (2004, [7]). The algebra βˆ— β„‚ (1949, [8]) was obtained by replacing the product of β„‚ with the one defined by π‘₯ ∘ 𝑦 = π‘₯ βˆ— 𝑦 βˆ— , where βˆ— means the standard involution of β„‚ .

We have the following very known results.

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

Proposition 2.2 (see [9]). If { π‘₯ 𝑖 } is a set of commuting elements in a flexible algebra 𝐴 over 𝐾 , then the subalgebra generated by the { π‘₯ 𝑖 } is commutative.

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

Lemma 2.4 (see [11]). Let 𝐴 = ( 𝑉 , ( β‹… , β‹… ) , Γ— ) be a quadratic algebra over 𝐾 . Then 𝐴 flexible if and only if ( β‹… , β‹… ) is symmetric and the following equivalent statements hold. (1) ( β‹… , β‹… ) is a trace form over 𝐴 .(2) ( β‹… , β‹… ) is a trace over 𝑉 .(3) ( 𝑒 Γ— 𝑣 , 𝑒 ) = 0 for every 𝑒 , 𝑣 ∈ 𝑉 .

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

We need the following results.

Theorem 2.6 (see [1]). Let 𝐴 be a real pre-Hilbert associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , or ℍ .

Theorem 2.7 (see [2]). Let 𝐴 be a real pre-Hilbert commutative algebra without divisors of zero and satisfying β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Suppose that 𝐴 containing a nonzero central element π‘Ž such that β€– π‘Ž π‘₯ β€– = β€– π‘Ž β€– β€– π‘₯ β€– for all π‘₯ ∈ 𝐴 . Then 𝐴 is isomorphic to ℝ , β„‚ , or βˆ— β„‚ .

Theorem 2.8 (see [1]). Let 𝐴 be a real pre-Hilbert alternative algebra with identity 𝑒 . Suppose that β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 and β€– 𝑒 β€– = 1 . Then 𝐴 is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 .

3. Real Pre-Hilbert Weakly Alternative Algebras

In this subparagraph, we prove that, if 𝐴 is a real pre-Hilbert algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional in the following cases. (1) 𝐴 is a real weakly alternative algebra without divisors of zero. (2) 𝐴 is a real Jordan algebra without divisors of zero. In the first case 𝐴 is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 , and 𝐴 is isomorphic to ℝ or β„‚ in the last case. Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 .

Lemma 3.1 (see [12]). Let 𝐴 be a real pre-Hilbert algebra with identity 𝑒 such that β€– π‘Ž 2 β€– = β€– π‘Ž β€– 2 for all π‘Ž ∈ 𝐴 and let 𝑉 = { π‘₯ ∈ 𝐴 / ( π‘₯ ∣ 𝑒 ) = 0 } then.(1) 𝑉 = { π‘₯ ∈ 𝐴 / π‘₯ 2 = βˆ’ β€– π‘₯ β€– 2 𝑒 } . (2) π‘₯ 𝑦 + 𝑦 π‘₯ = βˆ’ 2 ( π‘₯ ∣ 𝑦 ) 𝑒 for all π‘₯ , 𝑦 ∈ 𝑉 .

Remark 3.2. (i) The product π‘₯ ∧ 𝑦 = π‘₯ 𝑦 βˆ’ ( π‘₯ 𝑦 ∣ 𝑒 ) 𝑒 , for all π‘₯ , 𝑦 ∈ 𝑉 , provides 𝑉 the structure of an anticommutative algebra.
(ii) If 𝐴 is flexible, then ( π‘₯ 𝑦 ∣ 𝑒 ) = βˆ’ ( π‘₯ ∣ 𝑦 ) for all π‘₯ , 𝑦 ∈ 𝑉 .

Proof. (i) Let π‘₯ , 𝑦 ∈ 𝑉 , we have π‘₯ ∧ 𝑦 + 𝑦 ∧ π‘₯ = π‘₯ 𝑦 βˆ’ ( π‘₯ 𝑦 ∣ 𝑒 ) 𝑒 + 𝑦 π‘₯ βˆ’ ( 𝑦 π‘₯ ∣ 𝑒 ) 𝑒 = π‘₯ 𝑦 + 𝑦 π‘₯ βˆ’ ( π‘₯ 𝑦 + 𝑦 π‘₯ ∣ 𝑒 ) 𝑒 = βˆ’ 2 ( π‘₯ ∣ 𝑦 ) 𝑒 + 2 ( π‘₯ ∣ 𝑦 ) 𝑒 ( L e m m a 3 . 1 ) = 0 . ( 3 . 1 )
(ii) As 𝐴 is a flexible algebra, then = 0 = ( π‘₯ 𝑦 ) π‘₯ βˆ’ π‘₯ ( 𝑦 π‘₯ ) ( π‘₯ ∧ 𝑦 + ( π‘₯ 𝑦 ∣ 𝑒 ) 𝑒 ) π‘₯ βˆ’ π‘₯ ( 𝑦 ∧ π‘₯ + ( π‘₯ 𝑦 ∣ 𝑒 ) 𝑒 ) = ( π‘₯ ∧ 𝑦 ) π‘₯ + π‘₯ ( π‘₯ ∧ 𝑦 ) + ( ( π‘₯ 𝑦 ∣ 𝑒 ) βˆ’ ( 𝑦 π‘₯ ∣ 𝑒 ) ) π‘₯ = βˆ’ 2 ( π‘₯ ∣ π‘₯ ∧ 𝑦 ) 𝑒 + ( ( π‘₯ 𝑦 ∣ 𝑒 ) βˆ’ ( 𝑦 π‘₯ ∣ 𝑒 ) ) π‘₯ ( L e m m a 3 . 1 ) = ( ( π‘₯ 𝑦 ∣ 𝑒 ) βˆ’ ( 𝑦 π‘₯ ∣ 𝑒 ) ) π‘₯ . ( 3 . 2 ) This implies that ( π‘₯ 𝑦 ∣ 𝑒 ) = ( 𝑦 π‘₯ ∣ 𝑒 ) for all π‘₯ , 𝑦 ∈ 𝑉 , and by Lemma 3.1, we have ( π‘₯ 𝑦 + 𝑦 π‘₯ ∣ 𝑒 ) = βˆ’ 2 ( π‘₯ ∣ 𝑦 ) . Thus, ( π‘₯ 𝑦 ∣ 𝑒 ) = βˆ’ ( π‘₯ ∣ 𝑦 ) .

Theorem 3.3. Let 𝐴 be a real pre-Hilbert weakly alternative algebra with identity 𝑒 and without divisors of zero. Suppose that β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 and β€– 𝑒 β€– = 1 . Then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 .

Proof. It is sufficient to prove that 𝐴 is an alternative algebra.
Let π‘₯ , 𝑦 ∈ { 𝑒 } βŸ‚ such that ( π‘₯ ∣ 𝑦 ) = 0 , according to Lemma 3.1 we have π‘₯ 𝑦 + 𝑦 π‘₯ = 0 . ( 3 . 3 ) This implies that [ ] 0 = ( π‘₯ , π‘₯ , π‘₯ , 𝑦 ) = ( π‘₯ , π‘₯ , π‘₯ 𝑦 ) . ( 3 . 4 ) So π‘₯ [ π‘₯ ] ( π‘₯ 𝑦 ) = π‘₯ 2 ( π‘₯ 𝑦 ) = βˆ’ β€– π‘₯ β€– 2 π‘₯ 𝑦 . ( 3 . 5 ) As 𝐴 has nonzero divisors, then π‘₯ ( π‘₯ 𝑦 ) = βˆ’ β€– π‘₯ β€– 2 𝑦 = π‘₯ 2 𝑦 . ( 3 . 6 ) Therefore, ( π‘₯ , π‘₯ , 𝑦 ) = 0 . Now we take two arbitrary elements π‘₯ , 𝑦 ∈ { 𝑒 } βŸ‚ , and let 𝑧 = 𝑦 βˆ’ β€– π‘₯ β€– βˆ’ 2 ( π‘₯ ∣ 𝑦 ) π‘₯ ∈ { 𝑒 } βŸ‚ . Or ( π‘₯ ∣ 𝑧 ) = 0 , then ξ€· ( π‘₯ , π‘₯ , 𝑦 ) = π‘₯ , π‘₯ , 𝑧 + β€– π‘₯ β€– βˆ’ 2 ξ€Έ = ( π‘₯ ∣ 𝑦 ) π‘₯ ( π‘₯ , π‘₯ , 𝑧 ) = 0 . ( 3 . 7 ) Let π‘Ž = 𝛼 𝑒 + π‘₯ and 𝑏 = 𝛽 𝑒 + 𝑦 two elements in 𝐴 , with π‘₯ , 𝑦 ∈ { 𝑒 } βŸ‚ and 𝛼 , 𝛽 ∈ ℝ , we have ( π‘Ž βˆ’ 𝛼 𝑒 ) , ( 𝑏 βˆ’ 𝛽 𝑒 ) ∈ { 𝑒 } βŸ‚ . Therefore ( π‘Ž βˆ’ 𝛼 𝑒 , π‘Ž βˆ’ 𝛼 𝑒 , 𝑏 βˆ’ 𝛽 𝑒 ) = 0 , thus ( π‘Ž , π‘Ž , 𝑏 ) = 0 . So 𝐴 is a left alternative algebra. Now we show that 𝐴 is a right alternative algebra, if π‘₯ , 𝑦 ∈ { 𝑒 } βŸ‚ are two orthogonal elements. Then ξ€· ( π‘₯ 𝑦 ∣ π‘₯ ) = βˆ’ ( 𝑦 π‘₯ ∣ π‘₯ ) = βˆ’ 𝑦 ∣ π‘₯ 2 ξ€Έ = ( 𝑦 ∣ 𝑒 ) = 0 ( L e m m a 2 . 4 ) . ( 3 . 8 ) And ( π‘₯ 𝑦 ∣ 𝑒 ) = βˆ’ ( π‘₯ ∣ 𝑦 ) = 0 (Remark 3.2), thus, ( 𝑦 , π‘₯ , π‘₯ ) = ( 𝑦 π‘₯ ) π‘₯ βˆ’ 𝑦 π‘₯ 2 = βˆ’ π‘₯ ( 𝑦 π‘₯ ) + β€– π‘₯ β€– 2 𝑦 ( L e m m a 3 . 1 ) = π‘₯ ( π‘₯ 𝑦 ) βˆ’ π‘₯ 2 𝑦 = βˆ’ ( π‘₯ , π‘₯ , 𝑦 ) = 0 . ( 3 . 9 ) Similarly, we prove that ( 𝑏 , π‘Ž , π‘Ž ) = 0 for all π‘Ž , 𝑏 ∈ 𝐴 , then 𝐴 is a right alternative algebra. Thus, 𝐴 is an alternative algebra, the result ensues then of Theorem 2.8.

Corollary 3.4. Let 𝐴 be a real pre-Hilbert Jordan algebra with identity 𝑒 and without divisors of zero. Suppose that β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 and β€– 𝑒 β€– = 1 , then 𝐴 is finite dimensional and is isomorphic to ℝ or β„‚ .

Theorem 3.5. Let 𝐴 be a real pre-Hilbert weakly alternative algebra without divisors of zero. Suppose that β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 .

Proof. 𝐴 is a powers associative algebra (Proposition 2.3) then the subalgebra 𝐴 ( π‘₯ ) of 𝐴 , generated by π‘₯ ∈ 𝐴 , is associative and verifying the conditions of Theorem 2.6. Therefore, 𝐴 ( π‘₯ ) is isomorphic to ℝ or β„‚ , thus there is a nonzero idempotent 𝑒 ∈ 𝐴 such that π‘₯ 𝑒 = 𝑒 π‘₯ = π‘₯ ; that is, 𝐴 is a unital algebra of unit 𝑒 (Lemma 2.1). So the result is a consequence of Theorem 3.3.

Corollary 3.6. Let 𝐴 be a real pre-Hilbert Jordan algebra without divisors of zero. Suppose that β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to ℝ or β„‚ .

We give an extension of Theorem 3.3.

Theorem 3.7. Let 𝐴 be a real pre-Hilbert weakly alternative algebra without divisors of zero and satisfying β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Suppose that 𝐴 containing a nonzero central element π‘Ž such that β€– π‘Ž π‘₯ β€– = β€– π‘Ž β€– β€– π‘₯ β€– for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to ℝ , β„‚ , ℍ , or 𝕆 .

Proof . Let π‘₯ ∈ 𝐴 , the subalgebra 𝐴 ( π‘Ž , π‘₯ ) of 𝐴 generated by { π‘₯ , π‘Ž } is commutative. Theorem 2.7 implies that β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 , thus the result is a consequence of Theorem 3.5.

Corollary 3.8. Let 𝐴 be a real pre-Hilbert Jordan algebra without divisors of zero and satisfying β€– π‘₯ 2 β€– ≀ β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Suppose that 𝐴 contains a nonzero central element π‘Ž such that β€– π‘Ž π‘₯ β€– = β€– π‘Ž β€– β€– π‘₯ β€– for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to ℝ or β„‚ .

Remark 3.9. In the previous results the hypothesis without divisors of zero is necessary. The following example proves it.
Let 𝐻 be an infinite-dimensional real Hilbert space, we define the multiplication on the vector space 𝐴 = ℝ βŠ• 𝐻 by: ( 𝛼 + π‘₯ ) ( 𝛽 + 𝑦 ) = ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) + ( 𝛼 𝑦 + 𝛽 π‘₯ ) . ( 3 . 1 0 ) And the scalar product by ( ( 𝛼 + π‘₯ ) ∣ ( 𝛽 + 𝑦 ) ) = 𝛼 𝛽 + ( π‘₯ ∣ 𝑦 ) . ( 3 . 1 1 ) So 𝐴 is a commutative algebra satisfying β€– π‘Ž 2 β€– = β€– π‘Ž β€– 2 and ( π‘Ž 2 , 𝑏 , π‘Ž ) = 0 for all π‘Ž , 𝑏 ∈ 𝐴 . Indeed, we put π‘Ž = 𝛼 + π‘₯ and 𝑏 = 𝛽 + 𝑦 . We have β€– β€– ( 𝛼 + π‘₯ ) 2 β€– β€– 2 = β€– β€– ξ€· 𝛼 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ β€– β€– + 2 𝛼 π‘₯ 2 = ξ€· 𝛼 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ 2 + 4 𝛼 2 β€– π‘₯ β€– 2 = ξ€· 𝛼 2 + β€– π‘₯ β€– 2 ξ€Έ 2 = β€– 𝛼 + π‘₯ β€– 4 . ( 3 . 1 2 ) Then β€– π‘Ž 2 β€– = β€– π‘Ž β€– 2 , moreover, ξ€· π‘Ž 2 𝑏 ξ€Έ ξ€Ί π‘Ž = ( 𝛼 + π‘₯ ) 2 ξ€» = 𝛼 ( 𝛽 + 𝑦 ) ( 𝛼 + π‘₯ ) ξ€Ί ξ€· ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ξ€Έ ξ€» = 𝛼 + 2 𝛼 π‘₯ ( 𝛽 + 𝑦 ) ( 𝛼 + π‘₯ ) ξ€Ί ξ€· ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ξ€Έ + ξ€· ξ€· 𝛼 𝛽 βˆ’ 2 𝛼 ( π‘₯ 𝑦 ) 2 𝛼 𝛽 π‘₯ + 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ 𝑦 ξ€Έ ξ€» ( 𝛼 + π‘₯ ) . ( 3 . 1 3 ) Then ξ€· π‘Ž 2 𝑏 ξ€Έ 𝛼 π‘Ž = 𝛼 ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ξ€» βˆ’ ξ€Ί 𝛽 βˆ’ 2 𝛼 ( π‘₯ ∣ 𝑦 ) 2 𝛼 𝛽 β€– π‘₯ β€– 2 + ξ€· 𝛼 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ 𝑦 ξ€» + 𝛼 ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ξ€» ξ€Ί ξ€· 𝛼 𝛽 βˆ’ 2 𝛼 ( π‘₯ 𝑦 ) π‘₯ + 𝛼 2 𝛼 𝛽 π‘₯ + 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ 𝑦 ξ€» . ( 3 . 1 4 ) Thus, ξ€· π‘Ž 2 𝑏 ξ€Έ 𝛼 π‘Ž = ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) βˆ’ 2 𝛼 2 ( π‘₯ ∣ 𝑦 ) βˆ’ 2 𝛼 𝛽 β€– π‘₯ β€– 2 ξ€» + 𝛼 ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ( ξ€» . 𝛼 𝑦 + 𝛽 π‘₯ ) βˆ’ 2 𝛼 ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) π‘₯ ( 3 . 1 5 )
Similarly, π‘Ž 2 ( 𝑏 π‘Ž ) = ( 𝛼 + π‘₯ ) 2 [ ] = 𝛼 ( 𝛽 + 𝑦 ) ( 𝛼 + π‘₯ ) ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ξ€» [ ] . + 2 𝛼 π‘₯ ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) + ( 𝛼 𝑦 + 𝛽 π‘₯ ) ( 3 . 1 6 ) Thus, ξ€· π‘Ž 2 𝑏 ξ€Έ 𝛼 π‘Ž = ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) βˆ’ 2 𝛼 2 ( π‘₯ ∣ 𝑦 ) βˆ’ 2 𝛼 𝛽 β€– π‘₯ β€– 2 ξ€» + 𝛼 ξ€Ί ξ€· 2 βˆ’ β€– π‘₯ β€– 2 ξ€Έ ( ξ€» . 𝛼 𝑦 + 𝛽 π‘₯ ) βˆ’ 2 𝛼 ( 𝛼 𝛽 βˆ’ ( π‘₯ ∣ 𝑦 ) ) π‘₯ ( 3 . 1 7 )
From the two equalities (3.15) and (3.17), we conclude that ( π‘Ž 2 𝑏 ) π‘Ž = π‘Ž 2 ( 𝑏 π‘Ž ) ; that is, ( π‘Ž 2 , 𝑏 , π‘Ž ) = 0 for all π‘Ž , 𝑏 ∈ 𝐴 . This implies that 𝐴 is an infinite-dimensional real pre-Hilbert Jordan (weakly alternative) algebra with identity satisfying β€– π‘Ž 2 β€– = β€– π‘Ž β€– 2 and has a zero divisors. Indeed, let π‘₯ and 𝑦 be two orthogonal nonzero elements in 𝐻 , as defined multiplication of 𝐴 , we have π‘₯ 𝑦 = βˆ’ ( π‘₯ ∣ 𝑦 ) = 0 . Hence, 𝐴 is an algebra with zero divisors.

4. Complex Pre-Hilbert Noncommutative Jordan Algebras Satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2

We show that if 𝐴 is a noncommutative Jordan complex pre-Hilbert algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

4.1. Complex Pre-Hilbert Alternative Algebras Satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2

We need the following results.

Proposition 4.1 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to β„‚ .

Theorem 4.2 (see [3]). Let 𝐴 be a complex pre-Hilbert algebra with identity 𝑒 . Suppose that β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to β„‚ .

Lemma 4.3 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 has nonzero divisors.

Theorem 4.4 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative algebraic algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to β„‚ .

Lemma 4.5. Let 𝐴 be a complex pre-Hilbert alternative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 has nonzero divisors.

Proof. Let π‘Ž be a nonzero element in 𝐴 and let 𝑏 an element in 𝐴 such that π‘Ž 𝑏 = 0 . The subalgebra 𝐴 ( π‘Ž , 𝑏 ) of 𝐴 generated by { π‘Ž , 𝑏 } is associative (Theorem 2.5). We have β€– 𝑏 π‘Ž β€– 2 = β€– ( 𝑏 π‘Ž ) 2 β€– = β€– 𝑏 π‘Ž 𝑏 π‘Ž β€– = 0 then 𝑏 π‘Ž = π‘Ž 𝑏 = 0 . Thus, 𝐴 ( π‘Ž , 𝑏 ) is a commutative and associative, therefore, the Proposition 4.1 complete the demonstration.

Theorem 4.6. Let 𝐴 be a complex pre-Hilbert alternative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

Proof. Let π‘Ž ∈ 𝐴 , the subalgebra 𝐴 ( π‘Ž ) of 𝐴 generated by π‘Ž is commutative and associative (Theorem 2.5). Proposition 4.1 proves that 𝐴 ( π‘Ž ) is isomorphic to β„‚ , then there exists a nonzero idempotent 𝑓 ∈ 𝐴 . According to Theorem 4.2 it is sufficient to prove that 𝑓 is a unit element of 𝐴 . Let 𝑏 ∈ 𝐴 , we have 𝑓 ( 𝑏 βˆ’ 𝑓 𝑏 ) = 0 and ( 𝑏 βˆ’ 𝑏 𝑓 ) 𝑓 = 0 . As 𝐴 is without divisors of zero (Lemma 4.5), then 𝑓 𝑏 = 𝑏 𝑓 = 𝑏 . Thus, 𝐴 is finite dimensional and is isomorphic to β„‚ .

4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2

In this subparagraph we show that if ( 𝐴 , β€– β‹… β€– ) is a complex pre-Hilbert powers associative algebra (resp., flexible algebraic algebra, noncommutative Jordan algebra, or weakly alternative algebra) satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 is finite dimensional and is isomorphic to β„‚ .

We have the following importing result.

Lemma 4.7. Let 𝐴 be a complex pre-Hilbert powers associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 . Then 𝐴 has nonzero divisors.

Proof. Let π‘Ž be a nonzero element in 𝐴 , the subalgebra 𝐴 ( π‘Ž ) of 𝐴 is associative. According to Theorem 4.6, 𝐴 ( π‘Ž ) is isomorphic to β„‚ . Therefore, there exist a nonzero idempotent 𝑒 ∈ 𝐴 and 𝛼 ∈ ℝ βˆ’ { 0 } such that π‘Ž = 𝛼 𝑒 . Suppose there is a nonzero element 𝑏 ∈ { π‘Ž } βŸ‚ , as 𝐴 ( 𝑏 ) is isomorphic to β„‚ (Theorem 4.6), then there exist a nonzero idempotent 𝑓 ∈ 𝐴 and 𝛽 ∈ ℝ βˆ’ { 0 } such that 𝑏 = 𝛽 𝑓 . We have ( 𝑒 + 𝑓 ) 2 = 𝑒 + 𝑓 + 𝑒 𝑓 + 𝑓 𝑒 , and ( 𝑒 βˆ’ 𝑓 ) 2 = 𝑒 + 𝑓 βˆ’ 𝑒 𝑓 βˆ’ 𝑓 𝑒 = 2 ( 𝑒 + 𝑓 ) βˆ’ ( 𝑒 + 𝑓 ) 2 . ( 4 . 1 ) This implies that ( 𝑒 βˆ’ 𝑓 ) 2 ∈ 𝐴 ( 𝑒 + 𝑓 ) ∩ 𝐴 ( 𝑒 βˆ’ 𝑓 ) = { 0 } , because ( 𝑒 + 𝑓 ∣ 𝑒 βˆ’ 𝑓 ) = ( 𝑒 ∣ 𝑓 ) = 0 . Thus, ( 𝑒 βˆ’ 𝑓 ) 2 = 0 or β€– β€– 0 = ( 𝑒 βˆ’ 𝑓 ) 2 β€– β€– = β€– 𝑒 βˆ’ 𝑓 β€– 2 = 2 . ( 4 . 2 ) This is absurd and hence, 𝐴 has nonzero divisors.

Theorem 4.8. Let 𝐴 be a complex pre-Hilbert powers associative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

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

Theorem 4.9. Let 𝐴 be a complex pre-Hilbert flexible algebraic algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

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

We state the main theorem.

Theorem 4.10. Let 𝐴 be a complex pre-Hilbert noncommutative Jordan algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

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

Corollary 4.11. Let 𝐴 be a complex pre-Hilbert weakly alternative algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

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

Corollary 4.12. Let 𝐴 be a complex pre-Hilbert Jordan algebra satisfying β€– π‘₯ 2 β€– = β€– π‘₯ β€– 2 for all π‘₯ ∈ 𝐴 , then 𝐴 is finite dimensional and is isomorphic to β„‚ .

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

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

  1. B. Zalar, β€œOn Hilbert spaces with unital multiplication,” Proceedings of the American Mathematical Society, vol. 123, no. 5, pp. 1497–1501, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. A. Moutassim and A. Rochdi, β€œSur les algèbres préhilbertiennes vérifiant ∥x2∥≤∥x∥2,” Advances in Applied Clifford Algebras, vol. 18, no. 2, pp. 269–278, 2008. View at Publisher Β· View at Google Scholar
  3. M. R. Hilali, A. Moutassim, and A. Rochdi, β€œℂ-algebres normées préhilbertiennes vérifiant ∥x2∥=∥x∥2,” Advances in Applied Clifford Algebras, vol. 20, no. 1, pp. 33–41, 2010. View at Publisher Β· View at Google Scholar
  4. R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, NY, USA, 1966.
  5. A. A. Albert, β€œAbsolute valued real algebras,” Annals of Mathematics. Second Series, vol. 48, pp. 495–501, 1947. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch et al., Numbers, vol. 123 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1991.
  7. A. Rodriguez, β€œAbsolute-valued algebras, and absolute-valuable Banach spaces,” in Advanced Courses of Mathematical Analysis I, pp. 99–155, World Scientific, Hackensack, NJ, USA, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. A. A. Albert, β€œA note of correction,” Bulletin of the American Mathematical Society, vol. 55, p. 1191, 1949. View at Publisher Β· View at Google Scholar
  9. G. M. Benkart, D. J. Britten, and J. M. Osborn, β€œReal flexible division algebras,” Canadian Journal of Mathematics, vol. 34, no. 3, pp. 550–588, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. H. Braun and M. Koecher, Jordan-Algebren, Springer, Berlin, Germany, 1966.
  11. J. M. Osborn, β€œQuadratic division algebras,” Transactions of the American Mathematical Society, vol. 105, pp. 202–221, 1962. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. A. Moutassim, β€œQuelques résultats sur les algèbres flexibles préhilbertiennes sans diviseurs de zéro vérifiant ∥x2∥=∥x∥2,” Advances in Applied Clifford Algebras, vol. 18, no. 2, pp. 255–267, 2008. View at Publisher Β· View at Google Scholar