A Continuous Path of Singular Masas in the Hyperfinite II₁ Factor

Using methods of R.J.Tauer we exhibit an uncountable family of singular masas in the hyperfinite $\textrm{II}_1$ factor $\R$ all with Puk\'anszky invariant $\{1\}$, no pair of which are conjugate by an automorphism of $R$. This is done by introducing an invariant $\Gamma(A)$ for a masa $A$ in a \IIi factor $N$ as the maximal size of a projection $e\in A$ for which $A e$ contains non-trivial centralising sequences for $eN e$. The masas produced give rise to a continuous map from the interval $[0,1]$ into the singular masas in $\R$ equiped with the $d_{\infty,2}$-metric. A result is also given showing that the Puk\'anszky invariant is $d_{\infty,2}$-upper semi-continuous. As a consequence, the sets of masas with Puk\'anszky invariant $\{n\}$ are all closed.