Title: On smooth infinite dimensional grassmannians, splittings and non-commutative generalized cross-ratio mappings

Abstract: We describe basic diffeological structures related to splittings and Grassmannians for infinite dimensional vector spaces. We analyze and expand the notion of non-commutative cross-ratio and prove its smoothness. Then we illustrate this theory by examples, with some of them extracted from the existing literature related to infinite dimensional (Banach) Grassmannians, and others where the diffeological setting is a key primary step for rigorous definitions.