Spectral order unit spaces and JB-algebras

Order unit spaces with comparability and spectrality properties as introduced by Foulis are studied. We define continuous functional calculus for order unit spaces with the comparability property and Borel functional calculus for spectral order unit spaces. Applying the conditions of Alfsen and Schultz, we characterize order unit spaces with comparability property that are JB-algebras. We also prove a characterization of Rickart JB-algebras as those JB-algebras for which every maximal associative subalgebra is monotone $\sigma$-complete, extending an analogous result of Sait\^o and Wright for C*-algebras.