2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73782The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R-transform in this setting is given.16 pages. The revised version includes a new proof of additivity of the R-transform, as well as some minor correctionsOperator Algebras46L54text