All Questions
Tagged with oa.operator-algebras von-neumann-algebras
172 questions with no upvoted or accepted answers
28
votes
0
answers
2k
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Finite-dimensional subalgebras of $C^\star$-algebras
Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
- 25.5k
18
votes
0
answers
756
views
An "exercise" on von Neumann algebra tensor product
The following problem appears to be an easy exercise on von Neumann algebra tensor products, but since I've been failing to find a rigorous proof, I'd like to make sure it's not that trivial. Suppose $...
- 10.1k
15
votes
0
answers
790
views
Must we close weakly to apply the spectral theorem?
Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate.
The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
- 7,329
14
votes
0
answers
220
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Unitary group of a von Neumann algebra: is it a retract of $U(H)$?
Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$).
Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...
- 43.2k
14
votes
0
answers
647
views
Countably decomposable von Neumann algebras
A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
- 42.8k
10
votes
0
answers
426
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Twisted crossed product von Neumann Algebras
I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I ...
- 281
10
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0
answers
120
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Morita equivalence for graded von Neumann algebras
I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...
- 1,579
9
votes
0
answers
267
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Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?
Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map.
Can $T$ be extended to a `positively linear map' ...
- 11.3k
9
votes
0
answers
821
views
Pimsner-Popa Bases
Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} ...
- 5,425
8
votes
0
answers
252
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Question about the homogeneity of the state space of a type $\rm{III}_1$ factor
I'm reading the paper Homogeneity
of the State Space of Factors of Type $\rm{III}_1$ by Connes and Størmer.
Homogeneity of the state space means that all normal states are approximately unitarily ...
- 759
8
votes
0
answers
613
views
McDuff groups and McDuff factors
I asked a question over on Math.Stackexchange with the same title, but I didn't get any activity over there, which made me think that the question would be better suited for MathOverflow. I suppose ...
- 281
8
votes
0
answers
269
views
Shift on trivalent directed tree, operator and von Neumann algebra
Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
8
votes
0
answers
952
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About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
8
votes
0
answers
306
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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 ...
8
votes
0
answers
339
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Canonical Time Evolution for Type $II_{1}$-Factors?
This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
- 7,047
8
votes
0
answers
298
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Is the "Laplacian" a MASA in a Burnside Factor?
It is a basic fact that every type $II_{1}$ factor posesses a maximal abelian $*$-subalgebra (MASA). My question concerns the concrete realization of such subalgebras. For example, in the free group ...
- 7,047
8
votes
0
answers
345
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Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?
Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...
- 7,800
7
votes
0
answers
129
views
Strong contractibility of unitary group of properly infinite von Neumann algebras
In the introduction of their 1993 paper (see reference below), Popa and Takesaki write
As it turns out, in these topologies [the weak and strong topology] $U(\mathscr{H})$ is again contractible (cf.
[...
- 9,853
7
votes
0
answers
502
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Abstract characterization of group von Neumann algebra (II1 factor)
The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{...
7
votes
0
answers
268
views
Enveloping von Neumann algebra of Clifford algebra
As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra,...
- 9,853
7
votes
0
answers
127
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Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?
Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded ...
- 556
7
votes
0
answers
1k
views
Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
- 455
7
votes
0
answers
269
views
Approximation in the tensor square of a weakly exact von Neumann algebra
Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
- 25.8k
7
votes
0
answers
322
views
Is it known that "hyperfinite length" cannot distinguish free group factors?
Given a type $II_{1}$ factor $M$, Popa and Ge defined the hyperfinite length $l_{h}(M)$ of $M$ to be the minimum natural number $n$ such that there are hyperfinite subalgebras $R_{1}, R_{2},..., R_{n}$...
- 7,047
6
votes
0
answers
110
views
Standard form of fiber product of von Neumann algebras
Let $Z$ be an abelian von Neumann algebra, and let $A$ and $B$ be two von Neumann algebras that receive central maps $Z \to Z(A)$ and $Z \to Z(B)$.
We may then construct the fiber product of $A$ and $...
- 43.2k
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 ...
6
votes
0
answers
122
views
Premeasurability of affiliated operators for type $\textrm{III}$ von Neumann algebras
$\DeclareMathOperator\dom{dom}$If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action ...
- 7,047
6
votes
0
answers
378
views
What are some results that assume the Connes' embedding conjecture or any of its reformulations?
As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list):
...
- 265
6
votes
0
answers
232
views
Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
6
votes
0
answers
168
views
Characterizing fullness of a von Neumann algebra by the topology of its bimodules
Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
6
votes
0
answers
132
views
Schröder–Bernstein for representations of operator algebras
This is claimed in a Wikipedia Article:
If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
- 18.7k
6
votes
0
answers
441
views
Infinite number of non-isomorphic von Neumann algebras with property Gamma?
A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element $y\in\...
- 3,984
5
votes
0
answers
606
views
Weak Hopf algebra structure on twisted group algebra
A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map
$\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
- 393
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
0
answers
254
views
Examples of non-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors
Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\...
- 7,047
5
votes
0
answers
151
views
Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra
While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove:
Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
- 743
5
votes
0
answers
428
views
Koopman representation, weakly compact action, Ozawa Popa
Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
- 193
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 (...
5
votes
0
answers
241
views
Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$
In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
- 1,528
5
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0
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220
views
When can't spaces of correspondences distinguish type $II_{1}$ factors?
If $M$ is a type $II_{1}$ factor with trace $\tau$, let $Corr(M)$ denote the space of unitary equivalence classes of $M-M$ correspondences (binormal $M-M$ bimodules) equipped with Popa's analogue of ...
- 7,047
4
votes
0
answers
120
views
Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra
Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
- 113
4
votes
0
answers
115
views
Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$
I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...
- 131
4
votes
0
answers
123
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Extreme points in the space of ucp maps
Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
- 2,830
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 $\phi:\mathbb R\to Z(Out(R))$ ...
- 43.2k
4
votes
0
answers
253
views
Inflating the double dual of a C*-algebra (matrix algebra of double dual)
in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me ...
- 267
4
votes
0
answers
120
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On existence of property gamma of C star simple group von Neumann algebra
We call a finite vN algebra $M$ with trace $\tau$ has property Gamma iff there exist unitary $u_{n}\rightarrow 0$ weakly and $\|xu_{n}-u_{n}x\|\rightarrow 0$. My question can we have C*-simple ...
- 181
4
votes
0
answers
97
views
type III$_1$ states
Given a von Neumann algebra that is a type III$_1$ factor with the state $\omega$ and any $\epsilon>0$ is it always possible to find a projection or a partial isometry in the algebra such that its ...
4
votes
0
answers
131
views
Algebraic sum of relative commutants in a finite von Neumann algebra
Let $(M,\tau)$ be a finite von Neumann algebra with faithful normal tracial state $\tau$. Suppose that $A \subset M$ is a finite-dimensional abelian unital subalgebra--say $A = W^*(p_1,\dots, p_n)$ ...
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 ...
- 229
4
votes
0
answers
263
views
Approximately inner conditional expectations of $II_{1}$ factors
In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. ...
- 7,047