Groups associated to II1-factors☆

We extend recent work of the first named author, constructing a natural Hom semigroup associated to any pair of II1-factors. This semigroup always satisfies cancelation, hence embeds into its Grothendieck group. When the target is an ultraproduct of a McDuff factor (e.g., Rω), this Grothendieck group turns out to carry a natural vector space structure; in fact, it is a Banach space with natural actions of outer automorphism groups.