We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. We construct a highly indecomposable non-Gamma II ₁ factor N such that every separable von Neumann subalgebra of N with Haagerup’s property admits a unique embedding up to unitary conjugation. Such a factor necessarily has to be non-separable, but we show that it can be taken to have density character 2^₀. On the other hand, we are able to construct for any separable II ₁ factor M₀, a separable II ₁ factor M containing M₀ such that every property (T) subfactor admits a unique embedding into M up to uniformly approximate unitary equivalence; that is, any pair of embeddings can be conjugated up to a small uniform 2 -norm perturbation.
Gao et al. (Tue,) studied this question.