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

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