%% LyX 1.1 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[12pt,mathcal]{amsart} \usepackage{times} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \numberwithin{equation}{section} %% Comment out for sequentially-numbered \numberwithin{figure}{section} %% Comment out for sequentially-numbered \theoremstyle{plain} \newtheorem*{thm*}{Theorem} \theoremstyle{plain} \newtheorem*{cor*}{Corollary} \theoremstyle{plain} \newtheorem{lem}[thm]{Lemma} %%Delete [thm] to re-start numbering \theoremstyle{plain} \newtheorem{prop}[thm]{Proposition} %%Delete [thm] to re-start numbering \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \theoremstyle{remark} \newtheorem*{acknowledgement*}{Acknowledgement} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{amssymb} \usepackage{eepic} %\usepackage{pslatex} \usepackage[mathcal]{euscript} \def\id{\operatorname{id}} \def\diag{\operatorname{diag}} \evensidemargin 0in \oddsidemargin 0in \textwidth 6.5truein \topmargin 0.0truein \textheight 9truein \sloppy \makeatother \begin{document} \title{Irreducible subfactors of \protect\( L(\mathbb {F}_{\infty })\protect \) of index \protect\( \lambda >4\protect \).} \author{Dimitri Shlyakhtenko and Yoshimichi Ueda} \address{Department of Mathematics, UCLA, Los Angeles, CA 90095, USA.} \email{shlyakht@math.ucla.edu} \address{Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan.} \email{ueda@math.sci.hiroshima-u.ac.jp } \begin{abstract} By utilizing an irreducible inclusion of type III\( _{q^{2}} \) factors coming from a free-product type action of the quantum group \( SU_{q}(2) \), we show that the free group factor \( L(\mathbb {F}_{\infty }) \) possesses irreducible subfactors of arbitrary index \( >4 \). Combined with earlier results of Radulescu, this shows that \( L(\mathbb {F}_{\infty }) \) has irreducible subfactors with any index value in \( \{4\cos ^{2}(\pi /n):n\geq 3\}\cup [4,+\infty ) \). \end{abstract} \maketitle \section{Introduction.} Let \( M \) be a type II\( _{1} \) factor, and let \( N\subset M \) be a finite index subfactor. Recall that \( N\subset M \) is called \emph{irreducible,} if the relative commutant \( N'\cap M \) \emph{}is trivial. In his fundamental paper \cite{jones:index}, V.F.R.~Jones introduced the notion of index \( [M:N] \) of a subfactor \( N\subset M \). Related to the index, he introduced two invariants of a factor \( M \): \begin{eqnarray*} \mathcal{I}(M) & = & \{\lambda \in \mathbb {R}_{+}:\exists N\subset M\textrm{ subfactor},\, [M:N]=\lambda \},\\ \mathcal{C}(M) & = & \{\lambda \in \mathbb {R}_{+}:\exists N\subset M\textrm{ irreducible}\, \textrm{subfactor},\, [M:N]=\lambda \}, \end{eqnarray*} measuring the possible index values of subfactors of \( M \). In the same paper, Jones showed that for any factor \( M \), the index of a subfactor cannot be arbitrary; in fact, \( \mathcal{C}(M)\subset \mathcal{I}(M)\subset \mathcal{I}=\{4\cos ^{2}(\pi /n):n\geq 3\}\cup [4,+\infty ] \). Moreover, he produced subfactors of the hyperfinite II\( _{1} \) factor \( R \) of arbitrary allowable index, thus showing that \( \mathcal{I}(R)=\mathcal{I} \). However, the subfactors of \( R \) with index \( >4 \) that he constructed are not irreducible; to this day, it is still not known whether the equality \( \mathcal{C}(R)=\mathcal{I} \) holds; in other words, whether every index value can be realized by an irreducible subfactor of the hyperfinite factor. The one general positive result is due to by Popa \cite{popa:markov} (see also \cite{popa:universalconstructions} and \cite{popa:standardlattice}), stating that given a number \( \lambda \in \mathcal{I} \), there is a (non-hyperfinite) II\( _{1} \) factor \( M \), for which \( \lambda \in \mathcal{C}(M) \). Thanks to the theory of free probability, pioneered by Voiculescu in the early 80s (see e.g. \cite{DVV:book}), there has been much progress at understanding free group factors \( L(\mathbb {F}_{n}) \) (Voiculescu's introduction to \cite{DVV:book} states that these are the ``best'' of the ``bad'' non-hyperfinite factors). Following this philosophy, Radulescu undertook a study of subfactors of free group factors. Using a particular version of Popa's construction \cite{radulescu:subfact} and by employing free probability and random matrix techniques, he managed to show that for an interpolated free group factor \( L(\mathbb {F}_{t}) \), \( \mathcal{C}(L(\mathbb {F}_{t}))\cap [0,4]=\mathcal{I}(L(\mathbb {F}_{t}))\cap [0,4]=\mathcal{I}\cap [0,4] \). Coupled with his earlier result \cite{radulescu2} on the fundamental group of \( L(\mathbb {F}_{\infty }) \) this gave that \( L(\mathbb {F}_{\infty }) \) has (possibly non-irreducible) subfactors of all indices, i.e., \( \mathcal{I}(L(\mathbb {F}_{\infty }))=\mathcal{I} \). However, this still left the question about whether subfactors of \( L(\mathbb {F}_{\infty }) \) of index \( >4 \) can be chosen to be irreducible. The main theorem of this paper is \begin{thm*} For each \( \lambda >4 \), there exists an irreducible subfactor of \( L(\mathbb {F}_{\infty }) \) of index \( \lambda \). \end{thm*} Combined with earlier results of F. Radulescu, this gives: \begin{cor*} \( \mathcal{C}(L(\mathbb {F}_{\infty }))=\mathcal{I}(L(\mathbb {F}_{\infty }))=\mathcal{I} \). \end{cor*} In fact, more is true: given an allowable index value \( \lambda \in \mathcal{I} \), there exists an irreducible subfactor \( L(\mathbb {F}_{\infty })\cong N\subset M\cong L(\mathbb {F}_{\infty }) \) of index \( \lambda \). We would like to point out that it is still not known whether every Popa's \( \lambda \)-lattice \cite{popa:standardlattice} (i.e., an abstract system of higher relative commutant of a subfactor) can be realized by a subfactor of some fixed universal II\( _{1} \) factor \( M \). The factor \( M=L(\mathbb {F}_{\infty }) \) seems to be a likely candidate, although we don't know how to prove it; note, however, that Radulescu gave an affirmative answer to this question in the case that the \( \lambda \)-lattice is finite-depth. Also, by the results of this paper, all examples of irreducible subfactors whose \( \lambda \)-lattices come from the representation theory of \( SU_{q}(2) \) (see \cite{banica:compactQuantumGroupsSubfactors}, \cite{xu:latticesFromQuantumGroups}, \cite{wenzl:tensorCatergoriesQuantumGroups}) can be realized as \( N\subset M \) with \( N\cong M\cong L(\mathbb {F}_{\infty }) \). We don't know whether our result holds for \( L(\mathbb {F}_{n}) \), \( n<\infty \). The strategy to find irreducible subfactors of \( L(\mathbb {F}_{\infty }) \) with index \( \lambda >4 \) comes from the work of the second-named author \cite{ueda:fixed-point} on quantum \( SU_{q}(2) \) actions on free products of von Neumann algebras, and analysis of resulting Wassermann-type inclusions of type III factors. As was pointed out in \cite[Question 2]{ueda:fixed-point}, to prove the main theorem of the present paper, it is sufficient to identify a certain crossed product of a certain von Neumann algebra by \( SU_{q}(2) \), which we do in Section \ref{sec:amalgfreeprod} by identifying this algebra as a free Araki-Woods factor \cite{shlyakht:quasifree:big}. For the convenience of the reader, we summarize the main necessary facts from \cite{ueda:fixed-point} in Section \ref{sec:quantumactions}. \begin{acknowledgement*} The authors would like to express their sincere gratitude to MSRI and the organizers of the year-long program ``Operator Algebras (2000--2001)\char`\"{} held at MSRI for inviting them to the program, where this joint work (although conceived some time ago) was carried out and completed. D.S. would like to acknowledge support by a National Science Foundation Postdoctoral Fellowship. Y.U. would like to acknowledge support by the Japanese Ministry of Education, Science, Sports and Culture. \end{acknowledgement*} \section{Amalgamated free products of free group factors with \protect\( B(H)\protect \).\label{sec:amalgfreeprod}} In this section, we prove some technical results about amalgamated free products of free group factors with \( B(H) \), identifying them as free Araki-Woods factors. The treatment given in this section is more general than strictly speaking necessary for our result, but allows for more succinct and abstract proofs. We encourage the interested reader to look through the appendix to the paper, which can serve as an illustration of the proof given here. We recall some notation from \cite{shlyakht:semicirc}. Let \( M \) be a von Neumann algebra and \( \eta :M\to M\otimes B(\ell ^{2}) \) be a normal completely positive map, given as a matrix \( \eta (m)=(\eta _{ij}(m))_{ij} \). Let \( K \) be the \( M \)-Hilbert bimodule spanned by vectors \( \xi _{i} \), and satisfying \[ \langle \xi _{i},m\xi _{j}\rangle _{M}=\eta _{ij}(m),\quad \forall m\in M\] (see \cite[Lemma 2.2]{shlyakht:semicirc}). Let \( \mathcal{T} \) be the universal \( C^{*} \)-algebra generated by \( M \) and operators \( L(\zeta ):\zeta \in K \) satisfying \[ L(\zeta )^{*}mL(\zeta ')=\langle \zeta ,m\zeta '\rangle ,\quad \forall m\in M,\, \zeta ,\zeta '\in K.\] This is the Toeplitz extension of the Cuntz-Pimsner \( C^{*} \)-algebra associated to the \( M \)-bimodule \( H \) (see \cite{pimsner}). The operators \( L(\zeta ) \) are called \( \eta \)-creation operators over \( M \). Set \( L_{i}=L(\xi _{i}) \). Let \( E_{M} \) denote the canonical conditional expectation from \( \mathcal{T} \) onto \( M \) (see \cite{pimsner}, \cite{shlyakht:semicirc}). Choose a faithful normal state \( \phi _{0} \) on \( M \) and set \( \phi =\phi _{0}\circ E_{M} \). In the GNS representation associated to \( \phi \), consider the von Neumann algebra \[ \Phi (M,\eta )=W^{*}(M,L_{i}+L_{i}^{*}:i=1,2,\ldots ).\] It can be shown (\cite{shlyakht:semicirc}) that \( \Phi (M,\eta ) \) depends only on \( M \) and \( (\eta _{ij}) \). Let \( K^{\eta }\subset K \) be the \( \mathbb {R} \)-linear subspace of the Hilbert \( M \)-bimodule \( K \), given as the norm closure of the set of vectors \[ \sum _{\textrm{finite}}m_{j}\xi _{j}n^{*}_{j}+n_{j}\xi _{j}m_{j}^{*},\quad m_{j},n_{j}\in M.\] Then \cite[Proposition 2.19]{shlyakht:semicirc} \( \Phi (M,\eta ) \) depends only on the inclusion \( K^{\eta }\subset K \). In fact \begin{equation} \label{eqn:PhiandKeta} \Phi (M,\eta )=W^{*}(M,L(\zeta )+L(\zeta )^{*}:\zeta \in K^{\eta }\subset K) \end{equation} (in the GNS representation associated to \( \phi \)). In the particular case that \( M=\mathbb {C} \), the matrix \( \eta _{ij}(1) \) defines a scalar-valued inner product on \( K \), and \( K^{\eta }\subset K \) ends up a particular real subspace. In this case, under the assumption that the state \( E_{M}=E_{\mathbb {C}} \) is faithful, \( \Phi (M,\eta ) \) is nothing other than the free Araki-Woods factor \( \Gamma (K^{\eta }\subset K)'' \) in the notation of Remark 2.6 of \cite{shlyakht:quasifree:big} (see also \cite[Example 3.5]{shlyakht:semicirc}). A very particular example of such a matrix \( \mu \) is \[ \mu _{\lambda }=\left( \begin{array}{cc} 1 & -i\frac{\lambda -1}{\lambda +1}\\ i\frac{\lambda -1}{\lambda +1} & 1 \end{array}\right) ,\quad 0<\lambda <1;\] in this case \( \Phi (\mathbb {C},\mu _{\lambda }) \) is type III\( _{\lambda } \) and is the unique free Araki-Woods factor \( T_{\lambda } \) of this type. \begin{thm} \label{thrm:genresultA-valued}Let \( H \) be a finite or infinite dimensional Hilbert space. Let \( (\eta _{ij}):B(H)\to B(H)\otimes B(\ell ^{2}) \) be any normal completely positive map. Assume that \( E_{B(H)}:\Phi (B(H),\eta )\to B(H) \) is faithful. Then the von Neumann algebra \( \Phi (B(H),\eta ) \) is stably isomorphic to the von Neumann algebra \( \Phi (\mathbb {C},\mu ) \) for some (completely) positive map \( \mu :\mathbb {C}\to B(\ell ^{2}) \); moreover, \( E_{\mathbb {C}}:\Phi (\mathbb {C},\mu )\to \mathbb {C} \) is faithful. \end{thm} \begin{proof} By equation (\ref{eqn:PhiandKeta}), \[ N=\Phi (B(H),\eta )=W^{*}(B(H),L(\zeta )+L(\zeta )^{*}:\zeta \in K^{\eta }\subset K),\] where \( K \) and \( K^{\eta } \) are as described above. Let \( e_{ij} \), \( 0\leq i,j<\infty \) be a family of matrix units generating \( B(H) \); thus \[ e_{ij}e_{j'i'}=\delta _{jj'}e_{ii'},\quad e_{ij}^{*}=e_{ji}.\] Consider the algebra \[ P=e_{00}Ne_{00}.\] It is clear that \( P\otimes B(\ell ^{2})\cong N\otimes B(\ell ^{2}) \); thus it is sufficient to prove that \( P \) is isomorphic to \( \Phi (\mathbb {C},\mu ) \) for some \( \mu \). Note that \( P \) is generated by the family \[ e_{0i}(L(\zeta )+L(\zeta )^{*})e_{j0},\quad 0\leq i\leq j,\, \zeta \in K^{\eta };\] hence \[ P=W^{*}(L(e_{0i}\zeta e_{j0})+L^{*}(e_{0j}\zeta e_{i0}):\zeta \in K^{\eta }).\] Consider the linear space \begin{eqnarray*} V_{\mathbb {R}}=\textrm{span}_{\mathbb {R}}(\{e_{0i}\zeta e_{i0}:\zeta \in K^{\eta },i\geq 0\}\cup \{e_{0i}\zeta e_{j0}+e_{0j}\zeta e_{i0}:\zeta \in K^{\eta },0\leq i0)). \) Since \( M_{1} \) is a factor and \( L_{k} \) are \( \omega \)-creation operators over \( M \), where \( \omega _{ij}=\delta _{ij}C_{i}p_{i}\theta ((v^{*})^{i}dv^{i}), \) we get by \cite[Proposition 5.4]{shlyakht:amalg} that \( M\cong M_{1}*L(\mathbb {F}_{t}), \) where \( t=\sum _{i}\theta (v^{i}(v^{*})^{i})^{2} \). Hence \( M\cong L(\mathbb {F}_{\infty } \)). \end{proof} \section{Quantum \protect\( SU_{q}\protect \) actions.\label{sec:quantumactions}} The main goal of this section is to analyze inclusions of type III factors coming from a minimal free product type action of \( SU_{q}(2) \) on a von Neumann algebra. Let \( (A,\delta ) \) be the Hopf-von~Neumann algebra of Woronowicz's quantum group \( SU_{q}(2) \) \cite{woronowicz:SUq}, \cite{woronowicz:compactMatrixPseudogroups}, and denote by \( h \) the canonical Haar state on \( A \). Let \( V \) be the multiplicative unitary, i.e., the unitary on \( L^{2}(A)\otimes L^{2}(A) \) defined by \[ V(\Lambda _{h}(a)\otimes \zeta )=\delta (a)(\Lambda _{h}(1)\otimes \zeta ),\] and let \( W \) be the fundamental unitary, i.e., unitary on \( L^{2}(A)\otimes L^{2}(A) \) defined by \[ W(\eta \otimes \Lambda _{h}(a))=\delta (a)(\eta \otimes \Lambda _{h}(1)),\] where \( \Lambda _{h}:A\rightarrow L^{2}(A) \) is the canonical injection associated with the Haar state \( h \). The dual Hopf-von Neumann algebra \( \hat{A} \) is the von Neumann algebra acting on \( L^{2}(A) \) generated by elements of the form: \[ (\text {Id}\otimes \omega )(V),\quad \omega \in B(L^{2}(A))_{*}.\] Let \( m \) be an integer or \( +\infty \) and set \[ M=(L(\mathbb {F}_{m}),\tau )*(A,h)\] and let \( \Gamma :M\to M\otimes A \) be the free product of the trivial action of \( SU_{q}(2) \) on \( L(\mathbb {F}_{m}) \), and the left regular representation action \( \delta \) of \( SU_{q}(2) \) on \( A \) \cite[\S3]{ueda:fixed-point}. Recall that the crossed-product \( M\rtimes _{\Gamma }SU_{q}(2) \) is the von Neumann subalgebra of \( M\otimes B(L^{2}(A)) \) generated by \( \Gamma (M) \) and \( {\mathbb C}1\otimes \hat{A} \). For each irreducible finite-dimensional unitary representation \( \pi :V_{\pi }\to V_{\pi }\otimes A \) of \( SU_{q}(2) \), consider the Wassermann-type inclusion \cite{wassermann:ergodicActionsAndCoactionsSubfactors} of von Neumann algebras \[ \mathcal{N}=\mathbb {C}1\otimes M^{\Gamma }\subset (B(V_{\pi })\otimes M)^{\textrm{Ad}_{\pi }\otimes \Gamma }=\mathcal{M}.\] It turns out that the negative (normalized) \( q \)-trace \( \tau _{q}^{(\pi ,-)} \) gives rise to a finite-index conditional expectation \( \mathcal{E}:\mathcal{M}\to \mathcal{N} \) (see \cite[eq. (3.6)]{ueda:fixed-point}). We collect below some facts about this inclusion, most of which are from \cite{ueda:fixed-point}. \begin{thm} \label{thrm:propertiescrossedproduct} Let \( \pi \) be an irreducible representation of \( SU_{q}(2) \) and let \( \mathcal{N}\subset \mathcal{M} \) be as above, taken with the conditional expectation \( \mathcal{E} \). Then\\ (a) \( \mathcal{N} \) and \( \mathcal{M} \) are type III\( _{q^{2}} \) factors and \( \mathcal{N}\cong \mathcal{M} \);\\ (b) The inclusion \emph{\( \mathcal{N}\subset \mathcal{M} \)} is irreducible;\\ (c) The index of \( \mathcal{E} \) is the square of the \( q \)-dimension of \( \pi \), \( (\dim _{q}\pi )^{2} \);\\ (d) Let \( \phi \) denote the restriction of the free product state \( \tau *h \) to \( \mathcal{N}=M^{\Gamma }\subset M \). Then the centralizers \( \hat{\mathcal{N}}=\mathcal{N}^{\phi } \) and \( \hat{\mathcal{M}}=\mathcal{M}^{\phi \circ \mathcal{E}} \) are both factors of type II\( _{1} \);\\ (e) The inclusion \( \mathcal{N}\subset \mathcal{M} \) is ``essentially type II''. More precisely, the restriction of \( \mathcal{E} \) to the inclusion \( N\subset M \) is trace-preserving, and \[ \hat{\mathcal{N}}=\mathcal{N}^{\phi }\subset \mathcal{M}^{\phi \circ \mathcal{E}}=\hat{\mathcal{M}}\] is an inclusion of type II\( _{1} \) factors with the same index and the same system of higher relative commutants as \( \mathcal{N}\subset \mathcal{M} \). In particular, \( \hat{\mathcal{N}}\subset \hat{\mathcal{M}} \) is irreducible.\\ (f) The algebras \( \mathcal{N}=M^{\Gamma } \) and \( M\rtimes _{\Gamma }SU_{q}(2) \) are isomorphic. \end{thm} \begin{proof} The fact that \( \mathcal{N} \) is type III\( _{q^{2}} \) was proved in \cite{ueda:fixed-point}; this fact also follows from the explicit description of \( \mathcal{N} \) given in the appendix. For the convenience of the reader we give an expanded proof of the isomorphism \( \mathcal{N}\cong \mathcal{M} \) (see \cite[Remark 10]{ueda:fixed-point}). Let \( n \) be the dimension of \( V_{\pi } \) as a vector space. Choose a standard orthonormal system of vectors \( \xi _{i} \), \( 1\leq i\leq n \) for the space \( V_{\pi } \) so that the (co)representation \( \pi \) is given by \[ \pi (\xi _{j})=\sum _{i=1}^{n}\xi _{i}\otimes u_{ij}\in V_{\pi }\otimes A,\] for some \( u_{ij}\in A \) satisfying \( \sum _{k}u_{ki}^{*}u_{kj}=\delta _{ij} \). Since \( \mathcal{N}=M^{\Gamma }\subset M \) is type III, we can find isometries \( S_{i}\in M^{\Gamma } \), \( i=1,\dots ,n \) so that \( \sum S_{i}S_{i}^{*}=1 \) and \( S_{i}^{*}S_{j}=\delta _{ij} \). Set \[ v_{j}=\sum _{i=1}^{n}S_{i}\cdot u_{ij}\in M^{\Gamma }\cdot A\subset M,\quad H=\textrm{span}_{\mathbb {C}}\{v_{j}:1\leq j\leq n\}.\] Then \( v_{i}^{*}v_{j}=\delta _{ij} \) and \( \sum v_{i}v_{i}^{*}=1 \). Moreover, since \( S_{i}\in M^{\Gamma } \) are fixed by \( \Gamma \), we get (since \( \Gamma \) is an action and \( \Gamma |_{A}=\delta \)) \begin{eqnarray*} \Gamma (v_{j}) & = & \sum _{i}(S_{j}\otimes 1)\cdot \Gamma (u_{ij})\\ & = & \sum _{i}(S_{j}\otimes 1)\cdot \delta (u_{ij})\\ & = & \sum _{i,k}(S_{j}\otimes 1)\cdot (u_{ki}\otimes u_{ij})\\ & = & \sum _{i}V_{i}\otimes u_{ij}. \end{eqnarray*} It follows that the isomorphism \( H\ni v_{i}\mapsto \xi _{i}\in V_{\pi } \) is equivariant with respect to the restriction of \( \Gamma \) to \( H \) and the (co)representation \( \pi \) on \( V \). It follows that the isomorphism \[ \Phi :B(V_{\pi })\otimes M\to M,\quad \Phi ((q_{ij})_{ij})=\sum _{ij}v_{i}q_{ij}v_{j}^{*}\] is equivariant with respect to the action \( \textrm{Ad}\pi \otimes \Gamma \) on \( B(V_{\pi })\otimes M \) and the action \( \Gamma \) on \( M \). Thus \( \Phi \) restricts to an isomorphism of \( \mathcal{N}=M^{\Gamma } \) and \( \mathcal{M}=(B(V_{\pi })\otimes M)^{(\textrm{Ad}\pi \otimes \Gamma )} \). Thus (a) holds. Statements (b)--(e) were proved in \cite{ueda:fixed-point} (Theorem 8 and the discussion on pp. 43--49). For part (f), it is known that \[ M\rtimes _{\Gamma }SU_{q}(2)=(M\otimes B(L^{2}(A)))^{\widetilde{\Gamma }},\] where \( \widetilde{\Gamma }:=\text {Ad}(1\otimes \Sigma (W^{*}))\circ (\text {Id}\otimes \Sigma )\circ (\Gamma \otimes \text {Id}) \), and where \( \Sigma \) is the flip map (see \cite[Remark 20]{boca:ergodicActrionsQuantumGroups} for a simple proof, which also works in the \( W^{*} \)-algebraic setting without any change. For the reader's convenience, we should mention that the definition of ``\( \lambda (\omega ) \)'' in that paper involves a typographical error.) Since \( (M,\Gamma ) \) contains \( (A,\delta ) \), we see that \[ (M\otimes B(L^{2}(A)))^{\widetilde{\Gamma }}\cong M^{\Gamma }\otimes B(L^{2}(A))\] (see \cite[Lemma 4.2]{ueda:SUqActionFullFactor}) so that the crossed-product \( M\rtimes _{\Gamma }SU_{q}(2) \) and the fixed-point algebra \( M^{\Gamma } \) are isomorphic to each other, both being of type III. \end{proof} We would like to point out that the \( n \)-th step \( \mathcal{M}_{n} \) in the basic construction associated to \( \mathcal{N}\subset \mathcal{M} \) is rather easy to describe: \[ \mathcal{M}_{n}\cong (B(V_{\rho _{n}})\otimes M)^{\textrm{Ad}\rho _{n}\otimes \Gamma }\] where \begin{eqnarray*} \rho _{2s+1} & = & \hat{\pi }\otimes \rho _{2s}\\ \rho _{2s} & = & \pi \otimes \rho _{2s-1}\\ \rho _{0} & = & \pi , \end{eqnarray*} and \( \hat{\pi } \) is the canonical dual of \( \pi \). The conditional expectation \( \mathcal{E}_{n}:\mathcal{M}_{n}\to \mathcal{M}_{n-1} \) is given by \begin{eqnarray*} \mathcal{E}_{2s+1} & = & \tau _{q}^{(\hat{\pi },-)}\otimes \textrm{Id}\otimes \cdots \\ \mathcal{E}_{2s} & = & \tau _{q}^{(\pi ,-)}\otimes \textrm{Id}\otimes \cdots . \end{eqnarray*} This allows one to compute the \( \lambda \)-lattice of higher relative commutants of \( \mathcal{N}\subset \mathcal{M} \) (and thus of \( \hat{\mathcal{N}}\subset \hat{\mathcal{M}} \)) in terms of the representation theory of \( SU_{q}(2) \). In particular, one can show that if \( \pi \) is taken to be the fundamental representation, then the principal graph of \( \hat{\mathcal{N}}\subset \hat{\mathcal{M}} \) is \( A_{\infty } \), and the index is \( (q+q^{-1})^{2} \). For \( 0<\lambda <1 \), let \( T_{\lambda } \) denote the (unique) free Araki-Woods factor of type III\( _{\lambda } \) (see \cite{shlyakht:quasifree:big} and Section \ref{sec:amalgfreeprod}). \begin{prop} \label{prop:crossedproductisomTlambda}The factor \( ((L(\mathbb {F}_{\infty }),\tau )*(A,h))\rtimes _{\Gamma }SU_{q}(2) \) is isomorphic to the free Araki-Woods factor \( T_{q^{2}} \). Hence \( \mathcal{M}\cong \mathcal{N}\cong T_{q^{2}} \). \end{prop} \begin{proof} It is known that the fundamental unitary \( W \) lies in \( A\otimes \hat{A}' \) (acting on \( L^{2}(A)\otimes L^{2}(A) \)), and hence the mapping \( \Phi :x\in B(L^{2}(A))\mapsto W(1\otimes x)W^{*} \) gives rise to an isomorphism from \( B(L^{2}(A)) \) onto \( A\rtimes _{\delta }SU_{q}(2)=A\rtimes _{\delta }\hat{A} \) (see e.g. \cite{baaj-skandalis:multiplicativeUnitaries}). Moreover, the crossed product structure and the Haar state \( h \) on \( A \) give rise to a normal faithful conditional expectation \[ E:B(L^{2}(A))=A\rtimes _{\delta }\hat{A}\to \hat{A}\] given by \[ E(x)=(h\otimes \text {\textrm{Id}})(W(1\otimes x)W^{*}),\quad x\in B(L^{2}(A)).\] We claim that \[ M\rtimes _{\Gamma }SU_{q}(2)\cong (L(\mathbb {F}_{m})\otimes \hat{A},\tau \otimes \id )*_{\hat{A}}(B(L^{2}(A)),E).\] This isomorphism is obtained by extending the map \[ \Phi :A\rtimes _{\delta }SU_{q}(2)\to B(L^{2}(A))\] to \[ [(L(\mathbb {F}_{m}),\tau )*(A,h)]\rtimes _{\Gamma }SU_{q}(2)\supset A\rtimes _{\delta }SU_{q}(2)\] by mapping elements in \( L(\mathbb {F}_{m}) \) to the canonical copy of \( L(\mathbb {F}_{m}) \) inside \( (L(\mathbb {F}_{m})\otimes \hat{A},\tau \otimes \id )*_{\hat{A}}(B(L^{2}(A)),E) \). Since \( \Phi (\hat{a})=1\otimes \hat{a} \), \( \hat{a}\in \hat{A} \) and \( (h\otimes \text {Id})\circ \Phi =E \), \( \Phi \) extends to the desired isomorphism (a freeness condition needs to be checked, see \cite[Proposition 1]{ueda:fixed-point} and \cite{DVV:book}.) (If instead of a quantum group action, we would have an action of an ordinary group \( G \), the isomorphism above would be the content of the general identity \[ (R,\phi )*(S,\psi )\rtimes _{\textrm{id}*\alpha \textrm{ }}G=(R\otimes L(G),\phi \otimes \textrm{id})*_{L(G)}(S\rtimes _{\alpha }G,E_{G}),\] where \( E_{G} \) is the canonical \( L(G) \)-valued conditional expectation on \( S\rtimes _{\alpha }G \) associated to the state \( \psi :S\to \mathbb {C} \).) Applying Theorem \ref{thrm:freeproductBofH}, and noting that \( M\rtimes _{\Gamma }SU_{q}(2) \) is type III\( _{q^{2}} \) (Theorem \ref{thrm:propertiescrossedproduct} (f) and (a)), we find that \[ \mathcal{M}\cong \mathcal{N}\cong (L(\mathbb {F}_{m})\otimes \hat{A},\tau \otimes \id )*_{\hat{A}}(B(H),E)\] is isomorphic to the unique III\( _{q^{2}} \) free Araki-Woods factor \( T_{q^{2}} \). \end{proof} \begin{rem} Since the free Araki-Woods factor \( T_{q^{2}} \) is prime (i.e., cannot be written as a tensor product of two diffuse von Neumann algebras, see \cite{shlyakht:prime}), it follows that the inclusion \( \mathcal{N}\subset \mathcal{M} \) cannot be decomposed as a tensor product of a type II inclusion with a fixed type III factor, although the \( \lambda \)-lattices of the type III and type II inclusions coincide. (This remark can be also obtained from the results of Ge \cite{ge:entropy2}). \end{rem} \begin{lem} \label{lemma:centralizerLFinfty}With the assumptions of Theorem \ref{thrm:propertiescrossedproduct} (d), \( \hat{\mathcal{M}} \) and \( \hat{\mathcal{N}} \) are both isomorphic to \( L(\mathbb {F}_{\infty }) \). \end{lem} \begin{proof} Since \( \hat{N}=\mathcal{N}^{\phi } \) is a factor (Theorem \ref{thrm:propertiescrossedproduct} (d)) and \( \mathcal{N}=M^{\Gamma } \) is isomorphic to \( T_{q^{2}} \) , \( \hat{\mathcal{N}}\cong L(\mathbb {F}_{\infty }) \) (see Theorem \ref{thrm:coreofTlambda}). Similarly, since \( \mathcal{M}\cong T_{q^{2}} \) and \emph{\( \hat{\mathcal{M}}=\mathcal{M}^{\phi \circ \mathcal{E}} \)} is a factor, \( \hat{\mathcal{M}}\cong L(\mathbb {F}_{\infty }) \). \end{proof} For a general finite dimensional representation \( \pi \) the factor map structure associated with the type III inclusion \( \mathcal{M}\supseteq \mathcal{N} \) (see \cite{kosaki:typeIIIsubfactors}) can be described as follows. Decompose \[ \pi =n_{1}\cdot \pi _{\ell _{1}}\oplus \cdots \oplus n_{k}\cdot \pi _{\ell _{k}}\] into multiples of irreducible spin \( \ell \) representations \( \pi _{\ell } \) (see \cite{masudaANDCO:representationsSUQ2andQJacobiPolynomials}) with \( \ell _{i}\neq \ell _{j} \) (\( i\neq j \)). Set \( X:=\{1,2,\dots ,k\}\times [0,-\log q^{2}) \) and \( F_{t}(j,s):=(j,s+t) \), and let \( (X_{\mathcal{M}},F_{t}^{\mathcal{M}}) \) and \( (X_{\mathcal{N}},F_{t}^{\mathcal{N}}) \) be the flows of weights associated with \( \mathcal{M} \) and \( \mathcal{N} \), respectively, which are both identified with \( ([0,-\log q^{2}),\textrm{translation by }t) \). Let us define two factor maps \[ \pi ^{\mathcal{M}}:X\rightarrow X_{\mathcal{M}}=[0,-\log q^{2}),\quad \pi ^{\mathcal{N}}:X\rightarrow X_{\mathcal{N}}=[0,-\log q^{2})\] by \[ \pi ^{\mathcal{M}}(j,s)=s-\ell _{j}\log q^{2}\mod -\log q^{2},\quad \pi ^{\mathcal{N}}(j,s)=s.\] These \( \pi ^{\mathcal{M}} \), \( \pi ^{\mathcal{N}} \) describe the factor map structure. As a consequence, we get: \begin{rem} The inclusion \( \mathcal{M}\supseteq \mathcal{N} \) is of essentially type II if and only if all the set \[ \{\ell :\pi _{\ell }\textrm{ occurs in the decomposition of }\pi \}\] either consists entirely of half-integers or consists entirely of integers. Under this assumption, all statements in Theorem \ref{thrm:propertiescrossedproduct} still hold, with the exception of statement (b) (the resulting inclusion is of course no longer irreducible). \end{rem} This means that our method of constructing irreducible type II\( _{1} \) subfactors can be applied to the special standard \( \lambda \)-lattices arising from representations of the quantum group \( SU_{q}(2) \) (see \cite{banica:compactQuantumGroupsSubfactors}). We are now ready to prove our main result: \begin{thm} For every \( \lambda \in \mathcal{I} \), there exists an irreducible subfactor \( N \) of \( M=L(\mathbb {F}_{\infty }) \) of index \( \lambda \), so that \( N\cong L(\mathbb {F}_{\infty }) \). \end{thm} \begin{proof} The case \( \lambda \in \mathcal{I}\cap [0,4] \) was obtained by Radulescu in \cite{radulescu:subfact}. Let \( \lambda >4 \), and choose \( q \) so that \( \lambda =(q+q^{-1})^{2} \). Let \( \pi \) be the fundamental representation of \( SU_{q}(2) \). By Theorem \ref{thrm:propertiescrossedproduct} (e) and Lemma \ref{lemma:centralizerLFinfty}, for each \( 0