On conjugacy and perturbation of subalgebras

We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II₁ factor N₀ we construct a highly indecomposable non Gamma II₁ factor N such that N₀ ⊂ N and moreover every 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 of density character 2₀. On the other hand we are able to construct for any separable II₁ factor M₀, a separable II₁ factor M containing M₀ such that every property (T) subfactor admits a unique embedding into M up to uniformly approximate unitary equivalence; i.e., any pair of embeddings can be conjugated up to a small uniform $2$-norm perturbation.