### A Characterization of Sun-Reflexivity.

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We study the interplay between unconditional decompositions and the R-boundedness of collections of operators. In particular, we get several multiplier results of Marcinkiewicz type for ${L}^{p}$-spaces of functions with values in a Banach space X. Furthermore, we show connections between the above-mentioned properties and geometric properties of the Banach space X.

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria...

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