Derivations and Extensions in JC‐Algebras

A well‐known result by Upmeier states that every derivation on a universally reversible JC‐algebra A ⊆ B ( H ) sa extends to the C ∗ ‐algebra [ A ] generated by A in B ( H ). In this paper, we significantly strengthen this result by proving that every Jordan derivation on a universally reversible JC‐algebra A extends to ∗ ‐derivations on its universal enveloping real and complex C ∗ ‐algebras R ∗ ( A ) and C ∗ ( A ), respectively. Since every C ∗ ‐algebra generated by A is a quotient of its universal enveloping C ∗ ‐algebra C ∗ ( A ), Upmeier’s result follows as a direct corollary of our findings. Moreover, this work sheds new light on the deep structural relationship between JC‐algebras and their universal enveloping algebras, further enriching the understanding of their interplay. The proofs are rooted in the decomposition C ∗ ( A ) = R ∗ ( A ) ⊕ i R ∗ ( A ), where R ∗ ( A ) = { x ∈ C ∗ ( A ) : Φ A ( x ) = x ∗ } and Φ A is the unique ∗ ‐antiautomorphism of C ∗ ( A ) of order 2 that fixes the points of A .