Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space. Notice first that an associative algebra is a Jordan algebra if and only if it is commutative. To make it commutative we can define a new multiplication on $A$ to make it commutative, and in fact make it a Jordan algebra. The new multiplication $x ∘ y$ is the Jordan product:
$$(xy + yx)/2$$
This defines a unital Jordan algebra $A+$, and we call these unital Jordan algebras, as well as any unital subalgebras of these Jordan algebras, "special unital Jordan algebras".
Ok, now I was wondering:
Can all such special unital Jordan algebras be constructed and representated as above from some matrix algebra $B$ where the elements of the matrix are reals?
Does that matrix algebra $B$ always contain (non-zero) nilpotent elements $b^2 = 0$?
If 1) is 'no' ,does $A+$ always contain (non-zero) nilpotent elements $a^2 = 0$?
If 1) is 'yes', do $A+$ and $B$ both have elements that are square roots of $-1$ If at least one of them does?
An algebra over the real numbers is said to be formally real if it satisfies the property that a sum of $n$ squares can only vanish if each one vanishes individually.
If $A+$ is constructed from matrix algebra $B$ as described above, does that imply it is formally real?
I found this paper, not sure if it helps.
Maybe the ideas of tensors help me out, since they generalize matrices ?
Im no expert at this.
