All Questions
Tagged with subfactors von-neumann-algebras
72 questions
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91
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Subfactors with integer Jones index
Is there any integer (Jones) index subfactor which is not extremal?
- 393
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256
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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 ...
- 393
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111
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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 ...
- 393
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103
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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 ...
- 2,902
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118
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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 ...
- 393
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158
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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 ...
- 759
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71
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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 $\...
- 759
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87
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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'\...
- 759
1
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177
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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 ...
- 759
4
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110
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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 $\phi:\mathbb R\to Z(Out(R))$ ...
- 43.2k
6
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241
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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 ...
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138
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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 \...
5
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370
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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 ...
- 393
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399
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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 $M,e_P$...
- 393
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606
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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(...
- 393
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156
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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
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111
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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
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158
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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$-...
- 271
11
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490
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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 ...
- 43.2k
5
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119
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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-...
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75
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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 ...
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119
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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 ...
- 229
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122
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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
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1
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181
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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
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1
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187
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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$. ...
13
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349
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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 ...
- 5,425
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203
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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 ...
6
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1
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409
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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 $\...
- 547
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222
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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$ ...
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76
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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 ...
2
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101
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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....
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1
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122
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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 ...
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308
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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 ...
5
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161
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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 (...
3
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200
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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
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132
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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
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1
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190
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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) ...
2
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233
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${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?
Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{...
3
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2
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361
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${\rm II}_1$-factors with finite commutant and trivial intersection generate $B(H)$?
Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}'$, $...
3
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0
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156
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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$...
4
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1
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710
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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 ...
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3
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697
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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 ...
4
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2
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542
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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 ...
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2
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225
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Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
2
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0
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149
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Planar algebraic translation of a subfactor property
Let $N \subset M$ be an irreducible finite depth and finite index subfactor.
$M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows :
$$M=\bigoplus_{...
4
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1
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407
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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 \...
4
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0
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251
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An embedding theorem for a fusion ring planar algebra?
We first recall the embedding theorem for finite depth subfactor planar algebras:
The planar algebra generated by a (finite depth) subfactor, is embeddable into the planar algebra generated by its ...
3
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0
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304
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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 \...
3
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1
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335
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What's the natural equivalence of subfactors in general?
Let $A$ be a factor and $\mathcal{C}_{A}$ be the category of all the subfactors $(M \subset N)$ such that $M$ and $N$ are isomorphic to $A$. The most famous of them is perhaps $\mathcal{C}_{R}$ with $...
9
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3
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409
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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 ...