The Jordan multiplication semigroup of M_n(K)^+ equals the full End_K of the underlying vector space.
Lu,Additivity of Jordan maps on standard operator algebras, Linear Algebra Appl.357(2002), 123–131
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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.