Questions tagged [von-neumann-algebras]

Subtag of the [oa.operator-algebras] tag for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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Is this representation of Go (game) irreducible?

This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph. Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
Sebastien Palcoux's user avatar
27 votes
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2k views

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 $B \subset A$. ...
Andreas Thom's user avatar
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18 votes
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689 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 $...
Narutaka OZAWA's user avatar
18 votes
0 answers
888 views

local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...
André Henriques's user avatar
15 votes
0 answers
604 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; ...
Nik Weaver's user avatar
15 votes
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442 views

Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?

I recently gave a talk, where I talked about the tensor category of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor. Vaughan Jones, who was in the audience, later told me ...
André Henriques's user avatar
15 votes
0 answers
787 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 ...
Jonas Meyer's user avatar
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13 votes
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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 ...
André Henriques's user avatar
11 votes
0 answers
314 views

Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
Sebastien Palcoux's user avatar
10 votes
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376 views

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 ...
user193319's user avatar
9 votes
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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 ...
Anton Kapustin's user avatar
9 votes
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254 views

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' ...
Tomasz Kania's user avatar
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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 ...
Lau's user avatar
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*-homomorphisms from $L^\infty(0, 1)$ to itself that acts as the identity on continuous functions

Let $\pi: L^\infty([0, 1], \lambda) \rightarrow L^\infty([0, 1], \lambda)$ be a *-homomorphism (where $\lambda$ is the Lebesgue measure on $[0, 1]$), not necessarily normal (otherwise the question is ...
David Gao's user avatar
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8 votes
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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 ...
user193319's user avatar
8 votes
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257 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 $\...
Sebastien Palcoux's user avatar
8 votes
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927 views

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 ...
Sebastien Palcoux's user avatar
8 votes
0 answers
304 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 ...
Sebastien Palcoux's user avatar
8 votes
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337 views

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 (...
Jon Bannon's user avatar
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8 votes
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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 ...
Jon Bannon's user avatar
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8 votes
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799 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 $B_1=\{d b_{i_1} ...
Dave Penneys's user avatar
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Is there a finite-index finite-depth II$_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 < ...
Scott Morrison's user avatar
7 votes
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121 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. [...
Matthias Ludewig's user avatar
7 votes
0 answers
469 views

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{...
Sebastien Palcoux's user avatar
7 votes
0 answers
246 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,...
Matthias Ludewig's user avatar
7 votes
0 answers
119 views

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 ...
Peter's user avatar
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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 ...
Mathbuff's user avatar
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7 votes
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267 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 ...
Yemon Choi's user avatar
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7 votes
0 answers
315 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 $R_{1}, R_{2},..., R_{n}$...
Jon Bannon's user avatar
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6 votes
0 answers
232 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 ...
Sebastien Palcoux's user avatar
6 votes
0 answers
115 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 ...
Jon Bannon's user avatar
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6 votes
0 answers
366 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): ...
DUO Labs's user avatar
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6 votes
0 answers
224 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 ...
Matthias Ludewig's user avatar
6 votes
0 answers
156 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}(\...
Adrián González Pérez's user avatar
6 votes
0 answers
130 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 ...
Matthew Daws's user avatar
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6 votes
0 answers
426 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\...
Chris Ramsey's user avatar
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6 votes
0 answers
264 views

A conditional form of Holder's Inequality on Type II-1 von Neumann algebras

Suppose $V$ is a von-Neumann algebra with a trace $\tau$ so that $\tau(I) = 1$. Suppose $W$ is a sub-von Neumann algebra of $V$. Then we can define a conditional expection to be a projection $\...
Stephen Montgomery-Smith's user avatar
6 votes
0 answers
137 views

Is it true that there are exactly two conjugacy classes of order two elements in Out(R)?

In the title, $R$ stands for the hyperfinite III1 factor. An order two element $\alpha\in Out(M)$ ($M$ any factor) has an invariant $c(\alpha)\in H^3(\mathbb Z/2,S^1)=\mathbb Z/2$. Q: Is $c$ the ...
André Henriques's user avatar
5 votes
0 answers
595 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(...
Keshab Bakshi's user avatar
5 votes
0 answers
118 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-...
Sebastien Palcoux's user avatar
5 votes
0 answers
250 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 $\...
Jon Bannon's user avatar
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5 votes
0 answers
187 views

Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
truebaran's user avatar
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5 votes
0 answers
150 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 ...
John N.'s user avatar
  • 723
5 votes
0 answers
419 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. ...
Florin Radulescu's user avatar
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 (...
Sebastien Palcoux's user avatar
5 votes
0 answers
236 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 ...
Jiang's user avatar
  • 1,518
5 votes
0 answers
219 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 ...
Jon Bannon's user avatar
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4 votes
0 answers
121 views

Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?

Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group of automorphisms of $\...
Ruy's user avatar
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4 votes
0 answers
105 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 ...
kobeahibe's user avatar
4 votes
0 answers
115 views

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....
David Gao's user avatar
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