The Jordan multiplication semigroup of M_n(K)^+ equals the full End_K of the underlying vector space.
Jacobson,Lectures on Quadratic Jordan Algebras, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Vol
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The Jordan multiplication semigroup of matrix algebras is the full endomorphism semigroup
The Jordan multiplication semigroup of M_n(K)^+ equals the full End_K of the underlying vector space.