A Continuous Path of Singular Masas in the Hyperfinite II_1 Factor

Details A Continuous Path of Singular Masas in the Hyperfinite II_1 Factor A Continuous Path of Singular Masas in the Hyperfinite II_1 Factor

2006-02-08

arxiv.org/abs/math/0602155v1

J. London Math. Soc. (2) 75 (2007) 243-254

Mathematics

Operator Algebras

18 pages

Scientific paper

10.1112/jlms/jdl019

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.

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