There is no separable universal II1-factor
Abstract
Gromov constructed uncountably many pairwise non-isomorphic discrete groups with Kazhdan's property (T). We will show that no separable II1-factor can contain all these groups in its unitary group. In particular, no separable II1-factor can contain all separable II1-factors in it. We also show that the full group C*-algebras of some of these groups fail the lifting property.
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