Let $M$ be a II$_1$ factor von Neumann algebra with two-norm $\|\cdot\|_2$ induced by its canonical normal, faithful trace. It is said to be a McDuff Factor provided it tensorily absorbs the hyperfinite II$_1$ factor $R$; that is, $M \cong M \overline{\otimes} R$.
A sequence $(x_n)_{n \in \Bbb{N}} \in \ell^{\infty}(\Bbb{N},M)$ is said to be central if $$\lim_{n \to \infty}\|xx_n - x_n x\|_2 = 0$$ for any $x \in M$.Two central sequences $(x_n)_{n \in \Bbb{N}}$ and $(y_n)_{n \in \Bbb{N}}$ are said to be equivalent if $$\lim_{n \to \infty} \|x_n - y_n\|_2 =0$$ Finally, a central sequence is said to be trivial if it is equivalent to a scalar sequence.
I've also seen that $M$ is a McDuff factor is equivalent to the existence of pairs of non-commuting non-trivial central sequences in $M$. But I don't know exactly what that means. Would someone help me parse this?
Non-commuting central sequences means that you have two central sequences $(x_n)$, $(y_n)$ such that $(x_ny_n-y_nx_n)$ is not equivalent to the zero sequence. The point of view here is that one fixes a free ultrafilter $\omega$ and considers the II$_1$-factor $M^\omega=\ell^\infty(\mathbb N,M)/J$, where $J$ is the ideal of sequences equivalent to zero. The trace is $\tau_\omega([(x_n)])=\lim_\omega\tau(x_n)$. The (classes of) central sequences form $M'\cap M^\omega$. So two non-commuting central sequences correspond to two elements in $M'\cap M^\omega$ that do not commute.
McDuff proved that a II$_1$-factor $M$ is isomorphic to $M\otimes R$, with $R$ the hyperfinite II$_1$-factor, if and only if $M'\cap M^\omega$ is non-commutative; equivalently, if and only if there exist non-commuting central sequences in $M$.