On the existence of central sequences in subfactors

We prove a relative version of [Co1, Theorem 2.1] for a pair of type ${\text{I}}{{\text{I}}_1}$-factors $N \subset M$. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of $M$ contained in the subfactor $N$. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if $N$ has property $\Gamma$ and $G$ is an amenable group acting freely on $N$ via some action $\sigma$, then the crossed product $N{ \times _\sigma }G$ has property $\Gamma$. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems $1$, $2$ and $3$]), which gives necessary and sufficient conditions implying that the pair $N \subset M$ is stable.