Geometric and algebraic aspects of spectrality in order unit spaces: A comparison

Two approaches to spectral theory of order unit spaces are compared: the duality Alfsen and Shultz compression bases due Foulis. While former approach uses geometric properties an space in with a base norm space, latter notion is purely algebraic. It shown that Foulis strictly more general contains Alfsen-Shultz as special case. This demonstrated on two types examples: JB-algebras which if only they Rickart centrally symmetric state spaces, may be while not necessarily spectral.