In JB*-algebras without S_2(C) quotients, local quasi-linear Jordan functionals on self-adjoint elements are linear if and only if uniformly weakly continuous on the closed unit ball.
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The Quasi-linearity problem for Jordan-Banach algebras: a topological characterization
In JB*-algebras without S_2(C) quotients, local quasi-linear Jordan functionals on self-adjoint elements are linear if and only if uniformly weakly continuous on the closed unit ball.