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

Abstract

For an inclusion of the form $\mathbb{C}\subseteq M_n(\mathbb{C})$, where $M_n(\mathbb{C})$ is endowed with a state with diagonal weights $\lambda =({\lambda }_1,...,{\lambda }_n)$, we use Popa's construction, for non-tracial states, to obtain an irreducible inclusion of $I{I}_1$ factors, $N^{\lambda }(Q)\subseteq M^{\lambda }(Q)$ of index $\sum \frac{1}{{\lambda }_i}$. $M^{\lambda }(Q)$ is identified with a subfactor inside the centralizer algebra of the canonical free product state on $Q\star M_N(\mathbb{C})$. 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.

LaTeX2e amsart class; 17 pages (now single spaced); picture and minor corrections added