The Jordan multiplication semigroup of M_n(K)^+ equals the full End_K of the underlying vector space.
Dolinar,Maps on upper triangular matrices preserving Lie products, Linear Multilinear Algebra55 (2007), no
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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.