Abstract

Let 𝐴 be a real or complex algebra. Assuming that a vector space 𝐴 is endowed with a pre-Hilbert norm satisfying 𝑥2=𝑥2 for all 𝑥𝐴. 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 ,,, or 𝕆, and 𝐴 is isomorphic to in the last three cases. These last cases permit us to show that if 𝐴 is a complex pre-Hilbert noncommutative Jordan algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 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 𝑥2=𝑥2 for all 𝑥𝐴.

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 𝑥2𝑥2 for all 𝑥𝐴. Zalar (1995, [1]) proved that. (1)If 𝐴 is a real alternative algebra containing a unit element 𝑒 such that 𝑒=1, then 𝐴 is finite dimensional and is isomorphic to ,,, or 𝕆. (2)If 𝐴 is a real associative algebra satisfying 𝑥2=𝑥2, then 𝐴 is finite dimensional and is isomorphic to ,, or . (3)If 𝐴 is a complex normed algebra containing a unit element 𝑒 such that 𝑒=1, then 𝐴 is finite dimensional and is isomorphic to . These results were extended, respectively, to the following cases. (1)If 𝐴 is a real alternative algebra containing a nonzero central element 𝑎 such that 𝑎𝑥=𝑎𝑥, then 𝐴 is finite dimensional and is isomorphic to ,,, or 𝕆 (2008, [2]). (2)If 𝐴 is a real alternative algebra satisfying 𝑥2=𝑥2, then 𝐴 is finite dimensional and is isomorphic to ,, or (2008, [2]). (3)If 𝐴 is a complex normed algebra without divisors of zero and containing an invertible element 𝑣 such that 𝑣𝑥=𝑥𝑣=𝑣𝑥, then 𝐴 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 𝐴 is a real or complex pre-Hilbert algebra satisfying 𝑥2𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional in the following cases. (1)𝐴 is a real weakly alternative algebra without divisors of zero and satisfying 𝑥2=𝑥2 for all 𝑥𝐴 (Theorem 3.5). (2)𝐴 is a real weakly alternative algebra without divisors of zero and containing a nonzero central element 𝑎 such that 𝑎𝑥=𝑎𝑥 for all 𝑥𝐴 (Theorem 3.7). (3)𝐴 is a complex powers associative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴 (Theorem 4.8). In the first two cases 𝐴 is isomorphic to ,, or 𝕆 and 𝐴 is isomorphic to 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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 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 𝑥2=𝑥2 for all 𝑥𝐴.

2. Notation and Preliminary Results

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

Definitions 1. Let 𝐵 be an arbitrary algebra and 𝐾 is a field of characteristic not 2.(1)(i)𝐵 is called alternative if it is satisfied the identities (𝑦,𝑥,𝑥)=0 and (𝑥,𝑥,𝑦)=0 (where (,,) means associator), for all 𝑥,𝑦𝐵 (1966, [4]).(ii)𝐵 is called a powers associative if, for every 𝑥 in 𝐵, the subalgebra 𝐵(𝑥) generated by 𝑥 is associative.(iii) 𝐵 is called flexible if (𝑥,𝑦,𝑥)=0 for all 𝑥,𝑦𝐵.(iv)𝐵 is called a Jordan algebra if it is commutative and satisfied the Jordan identity: (J) (𝑥2,𝑦,𝑥)=0 for all 𝑥,𝑦𝐵.(v)𝐵 is called a noncommutative Jordan algebra if it is flexible and satisfied the Jordan identity (J).(vi)𝐵 is called weakly alternative if it is a noncommutative Jordan algebra and satisfied the identity (𝑥,𝑥,[𝑥,𝑦])=0 (where [,] means commutator). An alternative algebra or Jordan algebra is evidently weakly alternative.(vii)𝐵 is called quadratic if it has an identity element 𝑒 and satisfied the identity 𝑥2=𝛼𝑒+𝛽𝑥 for all 𝑥𝐵 and 𝛼,𝛽𝕂.(2)(viii) We say that 𝐵 is algebraic if, for every 𝑥 in 𝐵, the subalgebra 𝐵(𝑥) of 𝐵 generated by 𝑥 is finite dimensional (1947, [5]).(ix) A symmetric bilinear form (,) over 𝐵 is called a trace form if (𝑥𝑦,𝑧)=(𝑥,𝑦𝑧) for all 𝑥,𝑦,𝑧𝐵.(x)𝐵 is termed normed (resp., absolute valued) if it is endowed with a space norm such that 𝑥𝑦𝑥𝑦 (resp., 𝑥𝑦=𝑥𝑦), for all 𝑥,𝑦𝐵.(xi)𝐵 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 𝐵, a nonzero element which commute with all elements of the algebra 𝐵.
The most natural examples of absolute valued algebras are ,,, (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 (1949, [8]) was obtained by replacing the product of with the one defined by 𝑥𝑦=𝑥𝑦, where means the standard involution of .

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 {𝑥𝑖} is a set of commuting elements in a flexible algebra 𝐴 over 𝐾, then the subalgebra generated by the {𝑥𝑖} 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,(,),×) be a quadratic algebra over 𝐾. Then 𝐴 flexible if and only if (,) is symmetric and the following equivalent statements hold. (1)(,) is a trace form over 𝐴.(2)(,) is a trace over 𝑉.(3)(𝑢×𝑣,𝑢) = 0 for every 𝑢,𝑣𝑉.

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 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to ,, or .

Theorem 2.7 (see [2]). Let 𝐴 be a real pre-Hilbert commutative algebra without divisors of zero and satisfying 𝑥2𝑥2 for all 𝑥𝐴. Suppose that 𝐴 containing a nonzero central element 𝑎 such that 𝑎𝑥=𝑎𝑥 for all 𝑥𝐴. Then 𝐴 is isomorphic to ,, or .

Theorem 2.8 (see [1]). Let 𝐴 be a real pre-Hilbert alternative algebra with identity 𝑒. Suppose that 𝑥2𝑥2 for all 𝑥𝐴 and 𝑒=1. Then 𝐴 is isomorphic to ,,, or 𝕆.

3. Real Pre-Hilbert Weakly Alternative Algebras

In this subparagraph, we prove that, if 𝐴 is a real pre-Hilbert algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. 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 ,,, or 𝕆, and 𝐴 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 𝑥2=𝑥2 for all 𝑥𝐴.

Lemma 3.1 (see [12]). Let 𝐴 be a real pre-Hilbert algebra with identity 𝑒 such that 𝑎2=𝑎2 for all 𝑎𝐴 and let 𝑉={𝑥𝐴/(𝑥𝑒)=0} then.(1)𝑉={𝑥𝐴/𝑥2=𝑥2𝑒}. (2)𝑥𝑦+𝑦𝑥=2(𝑥𝑦)𝑒 for all 𝑥,𝑦𝑉.

Remark 3.2. (i) The product 𝑥𝑦=𝑥𝑦(𝑥𝑦𝑒)𝑒, for all 𝑥,𝑦𝑉, provides 𝑉 the structure of an anticommutative algebra.
(ii) If 𝐴 is flexible, then (𝑥𝑦𝑒)=(𝑥𝑦) for all 𝑥,𝑦𝑉.

Proof. (i) Let 𝑥,𝑦𝑉, we have 𝑥𝑦+𝑦𝑥=𝑥𝑦(𝑥𝑦𝑒)𝑒+𝑦𝑥(𝑦𝑥𝑒)𝑒=𝑥𝑦+𝑦𝑥(𝑥𝑦+𝑦𝑥𝑒)𝑒=2(𝑥𝑦)𝑒+2(𝑥𝑦)𝑒(Lemma3.1)=0.(3.1)
(ii) As 𝐴 is a flexible algebra, then =0=(𝑥𝑦)𝑥𝑥(𝑦𝑥)(𝑥𝑦+(𝑥𝑦𝑒)𝑒)𝑥𝑥(𝑦𝑥+(𝑥𝑦𝑒)𝑒)=(𝑥𝑦)𝑥+𝑥(𝑥𝑦)+((𝑥𝑦𝑒)(𝑦𝑥𝑒))𝑥=2(𝑥𝑥𝑦)𝑒+((𝑥𝑦𝑒)(𝑦𝑥𝑒))𝑥(Lemma3.1)=((𝑥𝑦𝑒)(𝑦𝑥𝑒))𝑥.(3.2) This implies that (𝑥𝑦𝑒)=(𝑦𝑥𝑒) for all 𝑥,𝑦𝑉, and by Lemma 3.1, we have (𝑥𝑦+𝑦𝑥𝑒)=2(𝑥𝑦). Thus, (𝑥𝑦𝑒)=(𝑥𝑦).

Theorem 3.3. Let 𝐴 be a real pre-Hilbert weakly alternative algebra with identity 𝑒 and without divisors of zero. Suppose that 𝑥2𝑥2 for all 𝑥𝐴 and 𝑒=1. Then 𝐴 is finite dimensional and is isomorphic to ,,, or 𝕆.

Proof. It is sufficient to prove that 𝐴 is an alternative algebra.
Let 𝑥,𝑦{𝑒} such that (𝑥𝑦)=0, according to Lemma 3.1 we have 𝑥𝑦+𝑦𝑥=0.(3.3) This implies that []0=(𝑥,𝑥,𝑥,𝑦)=(𝑥,𝑥,𝑥𝑦).(3.4) So 𝑥[𝑥](𝑥𝑦)=𝑥2(𝑥𝑦)=𝑥2𝑥𝑦.(3.5) As 𝐴 has nonzero divisors, then 𝑥(𝑥𝑦)=𝑥2𝑦=𝑥2𝑦.(3.6) Therefore, (𝑥,𝑥,𝑦)=0. Now we take two arbitrary elements 𝑥,𝑦{𝑒}, and let 𝑧=𝑦𝑥2(𝑥𝑦)𝑥{𝑒}. Or (𝑥𝑧)=0, then (𝑥,𝑥,𝑦)=𝑥,𝑥,𝑧+𝑥2=(𝑥𝑦)𝑥(𝑥,𝑥,𝑧)=0.(3.7) Let 𝑎=𝛼𝑒+𝑥 and 𝑏=𝛽𝑒+𝑦 two elements in 𝐴, with 𝑥,𝑦{𝑒} and 𝛼,𝛽, we have (𝑎𝛼𝑒),(𝑏𝛽𝑒){𝑒}. Therefore (𝑎𝛼𝑒,𝑎𝛼𝑒,𝑏𝛽𝑒)=0, thus (𝑎,𝑎,𝑏)=0. So 𝐴 is a left alternative algebra. Now we show that 𝐴 is a right alternative algebra, if 𝑥,𝑦{𝑒} are two orthogonal elements. Then (𝑥𝑦𝑥)=(𝑦𝑥𝑥)=𝑦𝑥2=(𝑦𝑒)=0(Lemma2.4).(3.8) And (𝑥𝑦𝑒)=(𝑥𝑦)=0 (Remark 3.2), thus, (𝑦,𝑥,𝑥)=(𝑦𝑥)𝑥𝑦𝑥2=𝑥(𝑦𝑥)+𝑥2𝑦(Lemma3.1)=𝑥(𝑥𝑦)𝑥2𝑦=(𝑥,𝑥,𝑦)=0.(3.9) Similarly, we prove that (𝑏,𝑎,𝑎)=0 for all 𝑎,𝑏𝐴, 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 𝑥2𝑥2 for all 𝑥𝐴 and 𝑒=1, then 𝐴 is finite dimensional and is isomorphic to or .

Theorem 3.5. Let 𝐴 be a real pre-Hilbert weakly alternative algebra without divisors of zero. Suppose that 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to ,,, or 𝕆.

Proof. 𝐴 is a powers associative algebra (Proposition 2.3) then the subalgebra 𝐴(𝑥) of 𝐴, generated by 𝑥𝐴, is associative and verifying the conditions of Theorem 2.6. Therefore, 𝐴(𝑥) is isomorphic to or , thus there is a nonzero idempotent 𝑒𝐴 such that 𝑥𝑒=𝑒𝑥=𝑥; 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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to or .

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 𝑥2𝑥2 for all 𝑥𝐴. Suppose that 𝐴 containing a nonzero central element 𝑎 such that 𝑎𝑥=𝑎𝑥 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to ,,, or 𝕆.

Proof . Let 𝑥𝐴, the subalgebra 𝐴(𝑎,𝑥) of 𝐴 generated by {𝑥,𝑎} is commutative. Theorem 2.7 implies that 𝑥2=𝑥2, 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 𝑥2𝑥2 for all 𝑥𝐴. Suppose that 𝐴 contains a nonzero central element 𝑎 such that 𝑎𝑥=𝑎𝑥 for all 𝑥𝐴. Then 𝐴 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 𝐻 be an infinite-dimensional real Hilbert space, we define the multiplication on the vector space 𝐴=𝐻 by: (𝛼+𝑥)(𝛽+𝑦)=(𝛼𝛽(𝑥𝑦))+(𝛼𝑦+𝛽𝑥).(3.10) And the scalar product by ((𝛼+𝑥)(𝛽+𝑦))=𝛼𝛽+(𝑥𝑦).(3.11) So 𝐴 is a commutative algebra satisfying 𝑎2=𝑎2 and (𝑎2,𝑏,𝑎)=0 for all 𝑎,𝑏𝐴. Indeed, we put 𝑎=𝛼+𝑥 and 𝑏=𝛽+𝑦. We have (𝛼+𝑥)22=𝛼2𝑥2+2𝛼𝑥2=𝛼2𝑥22+4𝛼2𝑥2=𝛼2+𝑥22=𝛼+𝑥4.(3.12) Then 𝑎2=𝑎2, moreover, 𝑎2𝑏𝑎=(𝛼+𝑥)2=𝛼(𝛽+𝑦)(𝛼+𝑥)2𝑥2=𝛼+2𝛼𝑥(𝛽+𝑦)(𝛼+𝑥)2𝑥2+𝛼𝛽2𝛼(𝑥𝑦)2𝛼𝛽𝑥+2𝑥2𝑦(𝛼+𝑥).(3.13) Then 𝑎2𝑏𝛼𝑎=𝛼2𝑥2𝛽2𝛼(𝑥𝑦)2𝛼𝛽𝑥2+𝛼2𝑥2𝑦+𝛼2𝑥2𝛼𝛽2𝛼(𝑥𝑦)𝑥+𝛼2𝛼𝛽𝑥+2𝑥2𝑦.(3.14) Thus, 𝑎2𝑏𝛼𝑎=2𝑥2(𝛼𝛽(𝑥𝑦))2𝛼2(𝑥𝑦)2𝛼𝛽𝑥2+𝛼2𝑥2(.𝛼𝑦+𝛽𝑥)2𝛼(𝛼𝛽(𝑥𝑦))𝑥(3.15)
Similarly, 𝑎2(𝑏𝑎)=(𝛼+𝑥)2[]=𝛼(𝛽+𝑦)(𝛼+𝑥)2𝑥2[].+2𝛼𝑥(𝛼𝛽(𝑥𝑦))+(𝛼𝑦+𝛽𝑥)(3.16) Thus, 𝑎2𝑏𝛼𝑎=2𝑥2(𝛼𝛽(𝑥𝑦))2𝛼2(𝑥𝑦)2𝛼𝛽𝑥2+𝛼2𝑥2(.𝛼𝑦+𝛽𝑥)2𝛼(𝛼𝛽(𝑥𝑦))𝑥(3.17)
From the two equalities (3.15) and (3.17), we conclude that (𝑎2𝑏)𝑎=𝑎2(𝑏𝑎); that is, (𝑎2,𝑏,𝑎)=0 for all 𝑎,𝑏𝐴. This implies that 𝐴 is an infinite-dimensional real pre-Hilbert Jordan (weakly alternative) algebra with identity satisfying 𝑎2=𝑎2 and has a zero divisors. Indeed, let 𝑥 and 𝑦 be two orthogonal nonzero elements in 𝐻, as defined multiplication of 𝐴, we have 𝑥𝑦=(𝑥𝑦)=0. Hence, 𝐴 is an algebra with zero divisors.

4. Complex Pre-Hilbert Noncommutative Jordan Algebras Satisfying 𝑥2=𝑥2

We show that if 𝐴 is a noncommutative Jordan complex pre-Hilbert algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

4.1. Complex Pre-Hilbert Alternative Algebras Satisfying 𝑥2=𝑥2

We need the following results.

Proposition 4.1 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative associative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to .

Theorem 4.2 (see [3]). Let 𝐴 be a complex pre-Hilbert algebra with identity 𝑒. Suppose that 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to .

Lemma 4.3 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 has nonzero divisors.

Theorem 4.4 (see [3]). Let 𝐴 be a complex pre-Hilbert commutative algebraic algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to .

Lemma 4.5. Let 𝐴 be a complex pre-Hilbert alternative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 has nonzero divisors.

Proof. Let 𝑎 be a nonzero element in 𝐴 and let 𝑏 an element in 𝐴 such that 𝑎𝑏=0. The subalgebra 𝐴(𝑎,𝑏) of 𝐴 generated by {𝑎,𝑏} is associative (Theorem 2.5). We have 𝑏𝑎2=(𝑏𝑎)2=𝑏𝑎𝑏𝑎=0 then 𝑏𝑎=𝑎𝑏=0. Thus, 𝐴(𝑎,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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

Proof. Let 𝑎𝐴, the subalgebra 𝐴(𝑎) of 𝐴 generated by 𝑎 is commutative and associative (Theorem 2.5). Proposition 4.1 proves that 𝐴(𝑎) is isomorphic to , then there exists a nonzero idempotent 𝑓𝐴. According to Theorem 4.2 it is sufficient to prove that 𝑓 is a unit element of 𝐴. Let 𝑏𝐴, we have 𝑓(𝑏𝑓𝑏)=0 and (𝑏𝑏𝑓)𝑓=0. As 𝐴 is without divisors of zero (Lemma 4.5), then 𝑓𝑏=𝑏𝑓=𝑏. Thus, 𝐴 is finite dimensional and is isomorphic to .

4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying 𝑥2=𝑥2

In this subparagraph we show that if (𝐴, ) is a complex pre-Hilbert powers associative algebra (resp., flexible algebraic algebra, noncommutative Jordan algebra, or weakly alternative algebra) satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 is finite dimensional and is isomorphic to .

We have the following importing result.

Lemma 4.7. Let 𝐴 be a complex pre-Hilbert powers associative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴. Then 𝐴 has nonzero divisors.

Proof. Let 𝑎 be a nonzero element in 𝐴, the subalgebra 𝐴(𝑎) of 𝐴 is associative. According to Theorem 4.6, 𝐴(𝑎) is isomorphic to . Therefore, there exist a nonzero idempotent 𝑒𝐴 and 𝛼{0} such that 𝑎=𝛼𝑒. Suppose there is a nonzero element 𝑏{𝑎}, as 𝐴(𝑏) is isomorphic to (Theorem 4.6), then there exist a nonzero idempotent 𝑓𝐴 and 𝛽{0} such that 𝑏=𝛽𝑓. We have (𝑒+𝑓)2=𝑒+𝑓+𝑒𝑓+𝑓𝑒, and (𝑒𝑓)2=𝑒+𝑓𝑒𝑓𝑓𝑒=2(𝑒+𝑓)(𝑒+𝑓)2.(4.1) This implies that (𝑒𝑓)2𝐴(𝑒+𝑓)𝐴(𝑒𝑓)={0}, because (𝑒+𝑓𝑒𝑓)=(𝑒𝑓)=0. Thus, (𝑒𝑓)2=0 or 0=(𝑒𝑓)2=𝑒𝑓2=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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

Proof. According to Lemma 4.7, 𝐴 has a nonzero divisors. Let 𝑎 be a nonzero element in 𝐴, then the subalgebra 𝐴(𝑎) of 𝐴 is associative. Theorem 4.6 implies that 𝐴(𝑎) is isomorphic to . 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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

Proof. Let 𝑎𝐴 be a nonzero element, according to Proposition 2.2 and Lemma 4.3, the subalgebra 𝐴(𝑎) of 𝐴 is commutative, algebraic, and without divisors of zero. Thus 𝐴(𝑎) is isomorphic to (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 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

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

Corollary 4.11. Let 𝐴 be a complex pre-Hilbert weakly alternative algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

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

Corollary 4.12. Let 𝐴 be a complex pre-Hilbert Jordan algebra satisfying 𝑥2=𝑥2 for all 𝑥𝐴, then 𝐴 is finite dimensional and is isomorphic to .

Proof. 𝐴 is a weakly alternative algebra. By Corollary 4.11, 𝐴 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.