A type ₁ factor with the smallest outer automorphism group

The canonical modular homomorphism provides an embedding of R \mathbb {R} into the outer automorphism group O u t ( M ) Out(M) of any type I I I 1 \mathrm {III}_{1} factor M M . We provide an explicit construction of a full factor of type I I I 1 \mathrm {III}_{1} with separable predual such that the outer automorphism group is minimal, i.e. this embedding is an isomorphism and a homeomorphism. We obtain such a I I I 1 \mathrm {III}_{1} factor by using an amalgamated free product construction.