# On Pre-Hilbert Noncommutative Jordan Algebras Satisfying∥x2∥=∥x∥2

Authors: Mohamed Benslimane; Abdelhadi Moutassim
Journal: ISRN Algebra
 (2012)

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## Abstract

LetA be a real or complex algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥·∥ satisfying ∥x2∥=∥x∥2 for all x∈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,ℂ,ℍ, or 𝕆, and A is isomorphic to ℂ 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∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ. Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying ∥x2∥=∥x∥2 for all x∈A.


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## Body

## 1. Introduction

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


## 2. Notation and Preliminary Results

Throughout the paper, the word algebra refers to a nonnecessarily associative algebra overℝ or ℂ.Definitions 1.
LetB 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,y∈B (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,y∈B.(iv)
B is called a Jordan algebra if it is commutative and satisfied the Jordan identity: (J) (x2,y,x)=0 for all x,y∈B.(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 x∈B and α,β∈𝕂.(2)
(viii)
We say thatB 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,z∈B.(x)
B is termed normed (resp., absolute valued) if it is endowed with a space norm ∥·∥ such that ∥xy∥≤∥x∥∥y∥ (resp., ∥xy∥=∥x∥∥y∥), for all x,y∈B.(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 inB, a nonzero element which commute with all elements of the algebra B.
The most natural examples of absolute valued algebras areℝ,ℂ,ℂ*,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 ℂ* (1949, [8]) was obtained by replacing the product of ℂ with the one defined by x∘y=x*y*, where * means the standard involution of ℂ.We have the following very known results.Lemma 2.1 (see [4]).
LetA 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]).
LetA be a noncommutative Jordan algebra over K, then A is a powers associative algebra.Lemma 2.4 (see [11]).
LetA=(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,v∈V.Theorem 2.5 (see [4]).
The subalgebra generated by any two elements of an alternative algebraA is associative.We need the following results.Theorem 2.6 (see [1]).
LetA be a real pre-Hilbert associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℝ,ℂ, or ℍ.Theorem 2.7 (see [2]).
LetA be a real pre-Hilbert commutative algebra without divisors of zero and satisfying ∥x2∥≤∥x∥2 for all x∈A. Suppose that A containing a nonzero central element a such that ∥ax∥=∥a∥∥x∥ for all x∈A. Then A is isomorphic to ℝ,ℂ, or ℂ*.Theorem 2.8 (see [1]).
LetA be a real pre-Hilbert alternative algebra with identity e. Suppose that ∥x2∥≤∥x∥2 for all x∈A and ∥e∥=1. Then A is isomorphic to ℝ,ℂ,ℍ, or 𝕆.


## 3. Real Pre-Hilbert Weakly Alternative Algebras

In this subparagraph, we prove that, ifA is a real pre-Hilbert algebra satisfying ∥x2∥=∥x∥2 for all x∈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,ℂ,ℍ, or 𝕆, and A is isomorphic to ℝ or ℂ 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∥=∥x∥2 for all x∈A.Lemma 3.1 (see [12]).
LetA be a real pre-Hilbert algebra with identity e such that ∥a2∥=∥a∥2 for all a∈A and let V={x∈A/(x∣e)=0} then.(1)
V={x∈A/x2=-∥x∥2e}.(2)
xy+yx=-2(x∣y)e for all x,y∈V.Remark 3.2.
(i) The productx∧y=xy-(xy∣e)e, for all x,y∈V, provides V the structure of an anticommutative algebra.
(ii) IfA is flexible, then (xy∣e)=-(x∣y) for all x,y∈V.Proof.
(i) Letx,y∈V, we have
(3.1)x∧y+y∧x=xy-(xy∣e)e+yx-(yx∣e)e=xy+yx-(xy+yx∣e)e=-2(x∣y)e+2(x∣y)e(Lemma3.1)=0.
(ii) AsA is a flexible algebra, then
(3.2)0=(xy)x-x(yx)=(x∧y+(xy∣e)e)x-x(y∧x+(xy∣e)e)=(x∧y)x+x(x∧y)+((xy∣e)-(yx∣e))x=-2(x∣x∧y)e+((xy∣e)-(yx∣e))x(Lemma3.1)=((xy∣e)-(yx∣e))x.
This implies that (xy∣e)=(yx∣e) for all x,y∈V, and by Lemma 3.1, we have (xy+yx∣e)=-2(x∣y). Thus, (xy∣e)=-(x∣y).Theorem 3.3.
LetA be a real pre-Hilbert weakly alternative algebra with identity e and without divisors of zero. Suppose that ∥x2∥≤∥x∥2 for all x∈A and ∥e∥=1. Then A is finite dimensional and is isomorphic to ℝ,ℂ,ℍ, or 𝕆.Proof.
It is sufficient to prove thatA is an alternative algebra.
Letx,y∈{e}⊥ such that (x∣y)=0, according to Lemma 3.1 we have
(3.3)xy+yx=0.
This implies that
(3.4)0=(x,x,[x,y])=(x,x,xy).
So
(3.5)x[x(xy)]=x2(xy)=-‖x‖2xy.
As A has nonzero divisors, then
(3.6)x(xy)=-‖x‖2y=x2y.
Therefore, (x,x,y)=0. Now we take two arbitrary elements x,y∈{e}⊥, and let z=y-∥x∥-2(x∣y)x∈{e}⊥. Or (x∣z)=0, then
(3.7)(x,x,y)=(x,x,z+‖x‖-2(x∣y)x)=(x,x,z)=0.
Let a=αe+x and b=βe+y two elements in A, with x,y∈{e}⊥ and α,β∈ℝ, we have (a-αe),(b-βe)∈{e}⊥. Therefore (a-αe,a-αe,b-βe)=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
(3.8)(xy∣x)=-(yx∣x)=-(y∣x2)=(y∣e)=0(Lemma2.4).
And (xy∣e)=-(x∣y)=0 (Remark 3.2), thus,
(3.9)(y,x,x)=(yx)x-yx2=-x(yx)+‖x‖2y(Lemma3.1)=x(xy)-x2y=-(x,x,y)=0.
Similarly, we prove that (b,a,a)=0 for all a,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.
LetA be a real pre-Hilbert Jordan algebra with identity e and without divisors of zero. Suppose that ∥x2∥≤∥x∥2 for all x∈A and ∥e∥=1, then A is finite dimensional and is isomorphic to ℝ or ℂ.Theorem 3.5.
LetA be a real pre-Hilbert weakly alternative algebra without divisors of zero. Suppose that ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℝ,ℂ,ℍ, or 𝕆.Proof.
A is a powers associative algebra (Proposition 2.3) then the subalgebra A(x) of A, generated by x∈A, is associative and verifying the conditions of Theorem 2.6. Therefore, A(x) is isomorphic to ℝ or ℂ, thus there is a nonzero idempotent e∈A 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.
LetA be a real pre-Hilbert Jordan algebra without divisors of zero. Suppose that ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℝ or ℂ.We give an extension of Theorem3.3.Theorem 3.7.
LetA be a real pre-Hilbert weakly alternative algebra without divisors of zero and satisfying ∥x2∥≤∥x∥2 for all x∈A. Suppose that A containing a nonzero central element a such that ∥ax∥=∥a∥∥x∥ for all x∈A. Then A is finite dimensional and is isomorphic to ℝ,ℂ,ℍ, or 𝕆.Proof .
Letx∈A, the subalgebra A(a,x) of A generated by {x,a} is commutative. Theorem 2.7 implies that ∥x2∥=∥x∥2, thus the result is a consequence of Theorem 3.5.Corollary 3.8.
LetA be a real pre-Hilbert Jordan algebra without divisors of zero and satisfying ∥x2∥≤∥x∥2 for all x∈A. Suppose that A contains a nonzero central element a such that ∥ax∥=∥a∥∥x∥ for all x∈A. Then A is finite dimensional and is isomorphic to ℝ or ℂ.Remark 3.9.
In the previous results the hypothesis without divisors of zero is necessary. The following example proves it.
LetH be an infinite-dimensional real Hilbert space, we define the multiplication on the vector space A=ℝ⊕H by:
(3.10)(α+x)(β+y)=(αβ-(x∣y))+(αy+βx).
And the scalar product by
(3.11)((α+x)∣(β+y))=αβ+(x∣y).
So A is a commutative algebra satisfying ∥a2∥=∥a∥2 and (a2,b,a)=0 for all a,b∈A. Indeed, we put a=α+x and b=β+y. We have
(3.12)‖(α+x)2‖2=‖(α2-‖x‖2)+2αx‖2=(α2-‖x‖2)2+4α2‖x‖2=(α2+‖x‖2)2=‖α+x‖4.
Then ∥a2∥=∥a∥2, moreover,
(3.13)(a2b)a=[(α+x)2(β+y)](α+x)=[((α2-‖x‖2)+2αx)(β+y)](α+x)=[((α2-‖x‖2)β-2α(xy))+(2αβx+(α2-‖x‖2)y)](α+x).
Then
(3.14)(a2b)a=α[(α2-‖x‖2)β-2α(x∣y)]-[2αβ‖x‖2+(α2-‖x‖2)y]α+[(α2-‖x‖2)β-2α(xy)]x+α[2αβx+(α2-‖x‖2)y].
Thus,
(3.15)(a2b)a=[(α2-‖x‖2)(αβ-(x∣y))-2α2(x∣y)-2αβ‖x‖2]a+[(α2-‖x‖2)(αy+βx)-2α(αβ-(x∣y))x].
Similarly,(3.16)a2(ba)=(α+x)2[(β+y)(α+x)]=[(α2-‖x‖2)+2αx][(αβ-(x∣y))+(αy+βx)].
Thus,
(3.17)(a2b)a=[(α2-‖x‖2)(αβ-(x∣y))-2α2(x∣y)-2αβ‖x‖2]a+[(α2-‖x‖2)(αy+βx)-2α(αβ-(x∣y))x].
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,b∈A. This implies that A is an infinite-dimensional real pre-Hilbert Jordan (weakly alternative) algebra with identity satisfying ∥a2∥=∥a∥2 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=-(x∣y)=0. Hence, A is an algebra with zero divisors.


## 4. Complex Pre-Hilbert Noncommutative Jordan Algebras Satisfying∥x2∥=∥x∥2

We show that ifA is a noncommutative Jordan complex pre-Hilbert algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.


### 4.1. Complex Pre-Hilbert Alternative Algebras Satisfying∥x2∥=∥x∥2

We need the following results.Proposition 4.1 (see [3]).
LetA be a complex pre-Hilbert commutative associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Theorem 4.2 (see [3]).
LetA be a complex pre-Hilbert algebra with identity e. Suppose that ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Lemma 4.3 (see [3]).
LetA be a complex pre-Hilbert commutative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Theorem 4.4 (see [3]).
LetA be a complex pre-Hilbert commutative algebraic algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Lemma 4.5.
LetA be a complex pre-Hilbert alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Proof.
Leta 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 ∥ba∥2=∥(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.
LetA be a complex pre-Hilbert alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Leta∈A, 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 ℂ, 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-fb)=0 and (b-bf)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 ℂ.


### 4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying∥x2∥=∥x∥2

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∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.We have the following importing result.Lemma 4.7.
LetA be a complex pre-Hilbert powers associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Proof.
Leta be a nonzero element in A, the subalgebra A(a) of A is associative. According to Theorem 4.6, A(a) is isomorphic to ℂ. Therefore, there exist a nonzero idempotent e∈A and α∈ℝ-{0} such that a=αe. Suppose there is a nonzero element b∈{a}⊥, as A(b) is isomorphic to ℂ (Theorem 4.6), then there exist a nonzero idempotent f∈A and β∈ℝ-{0} such that b=βf. We have (e+f)2=e+f+ef+fe, and
(4.1)(e-f)2=e+f-ef-fe=2(e+f)-(e+f)2.
This implies that (e-f)2∈A(e+f)∩A(e-f)={0}, because (e+f∣e-f)=(e∣f)=0. Thus, (e-f)2=0 or
(4.2)0=‖(e-f)2‖=‖e-f‖2=2.
This is absurd and hence, A has nonzero divisors.Theorem 4.8.
LetA be a complex pre-Hilbert powers associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
According to Lemma4.7, 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 ℂ. 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.
LetA be a complex pre-Hilbert flexible algebraic algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Leta∈A 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 ℂ (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.
LetA be a complex pre-Hilbert noncommutative Jordan algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Proposition2.3 implies that A is a powers associative algebra, and hence, A is isomorphic to ℂ (Theorem 4.8).Corollary 4.11.
LetA be a complex pre-Hilbert weakly alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
A is a noncommutative Jordan algebra. By Theorem 4.10, A is finite dimensional and is isomorphic to ℂ.Corollary 4.12.
LetA be a complex pre-Hilbert Jordan algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
A is a weakly alternative algebra. By Corollary 4.11, A is finite dimensional and is isomorphic to ℂ.


## 4.1. Complex Pre-Hilbert Alternative Algebras Satisfying∥x2∥=∥x∥2

We need the following results.Proposition 4.1 (see [3]).
LetA be a complex pre-Hilbert commutative associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Theorem 4.2 (see [3]).
LetA be a complex pre-Hilbert algebra with identity e. Suppose that ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Lemma 4.3 (see [3]).
LetA be a complex pre-Hilbert commutative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Theorem 4.4 (see [3]).
LetA be a complex pre-Hilbert commutative algebraic algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.Lemma 4.5.
LetA be a complex pre-Hilbert alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Proof.
Leta 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 ∥ba∥2=∥(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.
LetA be a complex pre-Hilbert alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Leta∈A, 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 ℂ, 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-fb)=0 and (b-bf)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 ℂ.


## 4.2. Complexes Pre-Hilbert Powers Associative Algebras Satisfying∥x2∥=∥x∥2

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∥=∥x∥2 for all x∈A. Then A is finite dimensional and is isomorphic to ℂ.We have the following importing result.Lemma 4.7.
LetA be a complex pre-Hilbert powers associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A. Then A has nonzero divisors.Proof.
Leta be a nonzero element in A, the subalgebra A(a) of A is associative. According to Theorem 4.6, A(a) is isomorphic to ℂ. Therefore, there exist a nonzero idempotent e∈A and α∈ℝ-{0} such that a=αe. Suppose there is a nonzero element b∈{a}⊥, as A(b) is isomorphic to ℂ (Theorem 4.6), then there exist a nonzero idempotent f∈A and β∈ℝ-{0} such that b=βf. We have (e+f)2=e+f+ef+fe, and
(4.1)(e-f)2=e+f-ef-fe=2(e+f)-(e+f)2.
This implies that (e-f)2∈A(e+f)∩A(e-f)={0}, because (e+f∣e-f)=(e∣f)=0. Thus, (e-f)2=0 or
(4.2)0=‖(e-f)2‖=‖e-f‖2=2.
This is absurd and hence, A has nonzero divisors.Theorem 4.8.
LetA be a complex pre-Hilbert powers associative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
According to Lemma4.7, 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 ℂ. 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.
LetA be a complex pre-Hilbert flexible algebraic algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Leta∈A 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 ℂ (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.
LetA be a complex pre-Hilbert noncommutative Jordan algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
Proposition2.3 implies that A is a powers associative algebra, and hence, A is isomorphic to ℂ (Theorem 4.8).Corollary 4.11.
LetA be a complex pre-Hilbert weakly alternative algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
A is a noncommutative Jordan algebra. By Theorem 4.10, A is finite dimensional and is isomorphic to ℂ.Corollary 4.12.
LetA be a complex pre-Hilbert Jordan algebra satisfying ∥x2∥=∥x∥2 for all x∈A, then A is finite dimensional and is isomorphic to ℂ.Proof.
A is a weakly alternative algebra. By Corollary 4.11, A is finite dimensional and is isomorphic to ℂ.



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