Let A be a Jordan algebra (with identity). If x is in A let A[x] be the subalgebra generated by x and the identity. An element x is regular if the dimension of A[x] is maximal.

For x regular, denote by tr(x) the trace of the operator "multiplication by x" restricted to A[x].

The function tr can be extended to all of A. This is explained for instance in the book by Faraut and Koranyi, "Analysis on symmetric cones".

My question is: how does one prove that this extended function tr is linear? (the point is to prove that it is additive, the homogeneity being easy)

In the book by Faraut and Koranyi, this seems to be assumed, there is no explanation. May be this is easy and I am missing something.

Ultimately, for instance if A is Euclidean and simple, one prove that tr is proportional to the function that maps x to the trace of the operator L(x) of multiplication by x (on all of A) which is clearly linear. But the proof of this last fact uses that tr is linear.

Thanks!

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