A spectral characterization of the socle of Banach Jordan systems (Q1358317)

Methods due to B. Aupetit are used to show that every properly spectrum-finite element \(x\) of a (complex) Banach-Jordan algebra or pair is spectrally bounded, i.e. there exists an integer \(m\) such that the spectrum of \(x\) in every homotope has at most \(m\) nonzero elements. If the Banach-Jordan algebra or pair has Jacobson radical zero, then the socle is precisely the set of properly spectrum-finite elements. This gives a positive answer to an open question put by O. Loos. Furthermore, in this case the algebraic rank of an element \(x\) (defined by O. Loos) agrees with the spectral rank of \(x\), that is, the smallest integer \(m\) as above.