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This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodým Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.

We establish that, in a universally complete complex K-space with a fixed multiplicative structure, the σ-distributivity of the base is equivalent to each of the following assertions: (1) every band preserving linear operator is order bounded; (2) there are no nonzero derivations; (3) every band preserving endomorphism is a band projection; (4) there are no nontrivial band preserving automorphisms.

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