  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.
