RO-algebras are defined and studied. For RO-algebra T , using properties of partial order, it is established that the set of bounded elements can be endowed with C-norm. The structure of commutative subalgebras of T is considered and the Spectral Theorem for any self-adjoint element of T is proven.

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

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