There is no separable universal 𝐼𝐼₁-factor

Abstract

Gromov constructed uncountably many pairwise nonisomorphic discrete groups with Kazhdan’s property ( T ) \mathrm {(T)} . We will show that no separable I I 1 \mathrm {II}_1 -factor can contain all these groups in its unitary group. In particular, no separable I I 1 \mathrm {II}_1 -factor can contain all separable I I 1 \mathrm {II}_1 -factors in it. We also show that the full group C ∗ C^* -algebras of some of these groups fail the lifting property.