Irreducible subfactors derived from Popa's construction for non-tracial states

Abstract

For an inclusion of the form $\Bbb C\subseteq M_n(\Bbb C)$, where $M_n(\Bbb C)$ is endowed with a state with diagonal weights $λ=(λ_1, ..., λ_n)$, we use Popa's construction, for non-tracial states, to obtain an irreducible inclusion of $II_1$ factors, $N^λ(Q)\subseteq M^λ(Q) $ of index $\sum \frac{1}{λ_i}$. $M^λ(Q)$ is identified with a subfactor inside the centralizer algebra of the canonical free product state on $Q\star M_N(\Bbb C)$. Its structure is described by ``infinite'' semicircular elements as in \cite{Ra3}. The irreducible subfactor inclusions obtained by this method are similar to the first irreducible subfactor inclusions, of index in $[4,\infty)$ constructed in \cite {Po1}, starting with the Jones' subfactors inclusion $R^s\subseteq R$, $s>4$. In the present paper, since the inclusion we start with has a simpler structure, it is easier to control the algebra structure of the subfactor inclusions.