Scalar characterization in Banach-Jordan algebras

Using a diagonalization Theorem obtained when the spectrum is Lipshitzian, we extend a result of G. Braatvedt on scalar characterization in Banach algebras to Banach-Jordan algebras. We also establish that any element of a semisimple Banach-Jordan algebra with the property that all elements in som neighbourhood of the identity are spectrally invariant under multiplication by the quadratic U operator, has analog with identity.