Groups associated to $II_1$-factors

We extend recent work of the first named author, constructing a natural Hom semigroup associated to any pair of II$_1$-factors. This semigroup always satisfies cancelation, hence embeds into its Grothendieck group. When the target is an ultraproduct of a McDuff factor (e.g., $R^ω$), 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.

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