On conjugacy and perturbation of subalgebras

Abstract

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 $2^{{\aleph }_0}$. 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; that is, any pair of embeddings can be conjugated up to a small uniform $2$-norm perturbation.