Can anyone please provide an example of a Jordan homomorphism (preferably on $n\times n$ matrices over a commutative ring) that is not already a homomorphism or antihomomorphism?
An obvious Jordan homomorphism is the transpose map, which is not a homomorphism but an antihomomorphism. A true example would be much appreciated.
A linear map $J:A \rightarrow B$ on two algebras $A$ and $B$ is called a Jordan homomorphism if $J(a^2) = J(a)^2$ for all $a \in A$. If $A$ and $B$ are unital and $1+1$ is invertible in $B$ then this is equivalent to $$J(ab+ba) = J(a)J(b)+J(b)J(a)$$ for all $a,b \in A$.