Spectral measures, boundedly 𝜎-complete Boolean algebras and applications to operator theory

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A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding L 1 {L^1} -spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek’s notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if B B is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade’s sense, then the closed operator algebras generated by B B with respect to the bounded convergence topology and the pointwise convergence topology coincide.