On conjugacy and perturbation of subalgebras

We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II${}_1$ factor $N_0$ we construct a highly indecomposable non Gamma II${}_1$ factor $N$ such that $N_0\subset N$ 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 $2^{{\mathrm{\aleph }}_0}$. On the other hand we are able to construct for any separable II${}_1$ factor $M_0$, a separable II${}_1$ factor $M$ containing $M_0$ 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.