Derivations and Extensions in JC-Algebras

Abstract

A well-known result by Upmeier states that every derivation on a universally reversible JC-algebra A⊆BHsa extends to the C∗-algebra A generated by A in BH. 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⊕iR∗A, where R∗A=x∈C∗A:ΦAx=x∗ and ΦA is the unique ∗-antiautomorphism of C∗A of order 2 that fixes the points of A.