Groups associated to $II_1$-factors

TL;DR

This paper constructs a Hom semigroup for pairs of II$_1$-factors, revealing algebraic and geometric structures such as a Banach space with automorphism group actions, extending previous work in operator algebras.

Contribution

It introduces a natural Hom semigroup associated to pairs of II$_1$-factors and explores its algebraic and geometric properties, including embedding into a Grothendieck group and vector space structure.

Findings

Semigroup always satisfies cancellation

Grothendieck group is a Banach space

Outer automorphism groups act naturally

Abstract

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., $R^\omega$), 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.