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1 vote
0 answers
91 views

Subfactors with integer Jones index

Is there any integer (Jones) index subfactor which is not extremal?
1 vote
1 answer
256 views

Intersection of two intermediate subalgebras

Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
3 votes
1 answer
103 views

Relation between factor condition on von Neumann algebras and modularity condition on ribbon fusion categories

A modular tensor category is defined to be a ribbon fusion category $\mathcal{C}$ in which the only objects which commute with all other objects are multiples of the identity. That is, if we denote by ...
1 vote
0 answers
111 views

Inclusion of finite dimensional C*-algebras and relative commutants of subfactors

Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...
3 votes
0 answers
200 views

What are the first non-maximal non-group-subgroup simple irreducible subfactors?

Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$ is normal if the biprojections $e_{\...
2 votes
0 answers
118 views

Depth of the reduced subfactor

Suppose $N\subset M$ is a finite depth subfactor with $[M:N]<\infty$. Consider the reduced subfactor $pNp\subset pMp$ for some projection $p\in N$. How to calculate the depth of $pNp\subset pMp$ in ...
4 votes
1 answer
710 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
7 votes
3 answers
697 views

Jordan-Hölder theorem for subfactors?

All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors. First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
3 votes
0 answers
304 views

Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?

The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
5 votes
1 answer
158 views

Backwards stable factors

A factor $R$ is called stable if $M_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor backwards stable if $R\cong M_n(S)$ implies $S\cong R$ where $S$ is allowed to be any ...
0 votes
1 answer
71 views

Cyclic vectors and subfactor inlcusion

Let $N\subset M$ be a be factors acting on a Hilbert space $H$. Denote by $\mathrm{Cyc}(A)\subset H$ the set of cyclic vectors for $A = M,N$. I am interested in the equality case of the inclusion $\...
17 votes
1 answer
2k views

The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results: Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980). A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-Popa ...
1 vote
0 answers
87 views

irreducible subfactor inclusion and commutativity of induced projections

Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$. Let $\Omega\in H$. Does it follow that the projections onto $[N\Omega]$ and $[M'\...
1 vote
1 answer
177 views

Commuting and generating subfactors of $ B(H)$

I have a question on subfactors of $B(H)$ (the von Neumann algebra of bounded operators on a complex Hilbert space). Say I have a subfactor $M$ of $B(H)$. Is it true that another subfactor $N \subset ...
4 votes
0 answers
110 views

non centrally free actions of ameanable groups on the hyperfinite III_1 factor

Let $R$ be a hyperfinite $\mathit{III}_1$ factor, and let $Out(R)$ be its set of automorphisms modulo inner automorphisms. There is a canonical and important homomorphism $\phi:\mathbb R\to Z(Out(R))$ ...
6 votes
0 answers
241 views

Tomita–Takesaki theory and subfactors

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $\Omega$ be a cyclic and separating vector in $H$. Let $J$ and $\Delta$ be the corresponding modular conjugation and modular ...
3 votes
0 answers
138 views

Is there a finite depth irreducible subfactor of prime index and not group-subgroup?

Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
0 votes
1 answer
122 views

On index 2 and square of subfactors without extra intermediate

Let $N \subsetneq K_i \subsetneq M$, $i=1,2$, be a square of irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, such that there is no extra intermediate, with $K_1 \not \...
1 vote
1 answer
181 views

Existence of a third intermediate if there are two intermediate subfactors of index 2

Let $(N \subset M)$ be an irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors. Let $K_1$ and $K_2$ be two distinct intermediate subfactors $N \subset K_i \subset M$, such ...
1 vote
0 answers
222 views

Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index (unital) inclusion of ${\rm II}_1$ factors. Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ ...
5 votes
0 answers
119 views

Pimsner-Popa basis dealing with higher relative commutants

Let $(N \subseteq M)$ be a finite index unital inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection. A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-...
5 votes
1 answer
203 views

What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth ...
13 votes
2 answers
582 views

Do subgroups have "two sided bases"?

Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of $$ E(g)=\begin{cases} g &\text{if } g\in H\\\ 0 &...
5 votes
1 answer
370 views

Action of a dual Hopf algebra on a factor

Suppose that a finite-dimesnional Hopf $C^*$-algebra $H$ acts on a type $II_1$ factor $N$ minimally (that is, $N^{\prime}\cap (N\rtimes H)=\mathbb{C}$). Is it true that there always exists a minimal ...
1 vote
0 answers
399 views

Pairs of subfactors

Suppose we have two subfactors $P\subset M$ and $Q\subset M$ with finite Jones indices (here $P,Q$ and $M$ all are $II_1$ factors). Under what condition the von Neumann algebra $L$ generated by $M,e_P$...
5 votes
0 answers
606 views

Weak Hopf algebra structure on twisted group algebra

A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map $\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
2 votes
0 answers
156 views

Extension of a theorem of Bisch to cyclotomic integers of fixed degree

Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...
3 votes
0 answers
111 views

Is there a non-irreducible maximal subfactor other than two-sided TLJ?

A subfactor $N \subseteq M$ is called: irreducible if $N' \cap M = \mathbb{C}$, maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$. The two-sided ...
3 votes
0 answers
158 views

Given non-type-I subfactors $R \subset S$, must $S$ have a projection that meets no projection in $R$ except $1$?

Let $R \subset S$ be distinct non-type-I von Neumann factors; say two projections $P, Q \in S$ "meet" if they have a common non-null subprojection (i.e. if $P \wedge Q \neq 0$), and call $P$ "$R$-...
11 votes
2 answers
490 views

Actions of locally compact groups on the hyperfinite $II_1$ factor

Let $R$ be the hyperfinite $II_1$ factor, and let $G$ be a locally compact group. (1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$? (2) If so, how does one ...
1 vote
0 answers
75 views

why can index be larger either the trace goes smaller or larger in index $\ge 4$ case?

I am studying the index of subfactor and basic construction recently. Suppose $M=\mathcal{R}$ to be the infinite hyperfinite II1 factor. For projection $p\in \mathcal{R}$, $p\mathcal{R}p$ is also a ...
13 votes
2 answers
349 views

Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?

Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...
4 votes
0 answers
119 views

Index of a subfactor of a full $II_1$ factor

On pg. 151 of "Coxeter Graphs and Towers of Algebras" by F.M. Goodman, P. de la Harpe, and V.F.R. Jones (1989), it is stated that there is no known example of a full $II_1$ factor having a subfactor ...
5 votes
1 answer
577 views

Is there a trivial construction of the trace on the Jones basic construction?

Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably ...
1 vote
1 answer
187 views

On the intersection of index 2 subfactors

Let $H_1$ and $H_2$ be two distinct index $2$ subgroups of a finite group $G$. We can deduce several properties about the intersection $H_1 \cap H_2$: $H_1$ and $H_2$ are normal subgroups of $G$. ...
9 votes
3 answers
409 views

What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors. Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
7 votes
2 answers
784 views

subfactor of finite rank but infinite index: is this possible?

A subfactor $N\subset M$ is essentially the same thing as an $N$-$M$-bimodule. I'll recall the basic definitions in the language of bimodules, and I hope that subfactor people will excuse me. ...
5 votes
0 answers
161 views

Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra? In other words, of the following form (...
6 votes
1 answer
409 views

Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states: Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on $\...
0 votes
0 answers
76 views

The completely reducible bimodules coming from subfactors

This post is a sequel of: Are all the R-R-bimodules completely reducible? Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
8 votes
0 answers
306 views

Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $. Question: Are there only finitely many maximal irreducible amenable subfactors at ...
2 votes
0 answers
101 views

Are there infinitely many amenable Hadamard-Petrescu subfactors?

The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by W....
0 votes
1 answer
122 views

Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here). Now ...
3 votes
0 answers
156 views

A section from subfactors to transitive groups

A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite II$...
15 votes
1 answer
547 views

Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor

Let $R$ be the hyperfinite type $III_1$ factor, and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology (topology of pointwise convergence on the predual). An automorphism $\...
4 votes
1 answer
407 views

Are all the R-R-bimodules completely reducible?

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule. Question: Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)? Example: If $(N \...
1 vote
0 answers
308 views

Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?

Let $(N \subset M)$ be a finite depth-index irreducible subfactor. Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor? (In others words, is ...
4 votes
2 answers
542 views

The category of subfactors extending the category of groups?

This post was inspired by this answer of Dave Penneys. In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
2 votes
0 answers
132 views

Uniqueness of the tensor product decomposition of subfactors

A subfactor $(N \subset M)$ is indecomposable if (for $N_i \subset M_i)$: $$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$ Then, a subfactor $(N ...
1 vote
1 answer
190 views

Infinite amenable group subfactors

Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$. Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...