All Questions
Tagged with von-neumann-algebras gr.group-theory
39 questions
5
votes
1
answer
250
views
Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?
Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...
- 525
3
votes
1
answer
285
views
Extreme points of the set of all traces
Let $G$ be a finitely generated group with a bound on its complex unitary irreducible representations: That is assume all complex unitary irreducibles of $G$ have degrees at most $k$ for some integer $...
- 193
5
votes
1
answer
341
views
The center of a representation von Neumann algebra, and finite index subgroups
Consider a (countable) group $G$, a subgroup $H\leq G$ of finite index, and a unitary representation $\pi:G\to \mathcal{U}(\mathcal{H})$.
If the center of the von Neumann algebra $\pi(H)''$ is finite ...
10
votes
0
answers
426
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 ...
- 281
7
votes
1
answer
403
views
Induction and restriction of unitary representations
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$,
let $\Rep(G)$ and $\Rep(H)$ denote their ...
- 43.2k
9
votes
1
answer
372
views
Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?
Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
- 741
8
votes
0
answers
614
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
2
votes
0
answers
99
views
Convergence of Brown measures
For each $n\in \mathbb N$, let $\mathcal M_n$ be a finite von Neumann algebra with a faithful trace $\tau_n$. Fix a non-principal ultrafilter $\omega$ on $\mathbb N$. Let $\mathcal M^\omega$ be the ...
- 1,418
5
votes
2
answers
1k
views
Unusual crossed product constructions being factors
Let $A$ be an abelian von Neumann algebra and $G$ a countable group acting on $A$. In the literature we meet usually two kinds of crossed product $A \rtimes G$ being a factor:
if the action is (...
7
votes
0
answers
502
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{...
0
votes
1
answer
204
views
A certain class of representations
Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?
(The word "finite-dimensional" was ...
- 2,118
2
votes
0
answers
96
views
Could we assume without loss of generality that all coefficients are positive?
Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
- 2,118
3
votes
1
answer
388
views
Fundamental group and group measure space construction
Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
5
votes
1
answer
306
views
Cartan subalgebra and group measure space construction
Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
1
vote
0
answers
185
views
Unitary element of the group algebra
Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
- 2,118
3
votes
0
answers
109
views
Does this element belong to $\mathbb CG$?
Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
- 2,118
2
votes
0
answers
201
views
An example of non trivial projections in a group von Neumann algebra
Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
- 2,118
3
votes
1
answer
192
views
An analytical zero divisor
Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
- 2,118
3
votes
0
answers
206
views
Do these limits exist?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is ...
- 2,118
4
votes
1
answer
229
views
About some positive elements in a group von Neumann algebra
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^*$ is defined to ...
- 2,118
2
votes
1
answer
208
views
an example of discrete factor group of exponential growth
I would like to understand if there is a discrete infinite group of exponential growth/intermediate growth such that its group von Neumann algebra is a $II_1$ factor. I would be happy to get an ...
- 361
4
votes
1
answer
106
views
Does the inclusion of the discretized group into itself lift to the group von Neumann algebras?
Let $G$ be a locally compact, Hausdorff and $2^{nd}$-countable group and let $G_{disc}$ be the same group with the discrete topology. We have a continuous (and bijective) homomorphism given by
$$
...
- 1,090
4
votes
1
answer
325
views
Do irreducible characters form a closed set?
A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive
type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A ...
- 1,418
11
votes
1
answer
635
views
Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space
Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$
and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that $\VN(\...
- 25.8k
4
votes
1
answer
1k
views
Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be (...
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: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...
4
votes
1
answer
710
views
Abelian subfactors, a relevant concept?
Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
7
votes
3
answers
698
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 ...
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 ...
0
votes
2
answers
225
views
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 $II_1$ ...
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
23
votes
4
answers
2k
views
Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
- 9,486
9
votes
1
answer
1k
views
Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,057
8
votes
0
answers
298
views
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 ...
- 7,057
12
votes
2
answers
1k
views
Do Burnside Group Factors have Gamma?
The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so ...
- 7,057
5
votes
2
answers
670
views
Reference for von Neumann algebras coming from a group algebra twisted by a 2-cocycle?
I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, ...
- 363
8
votes
3
answers
646
views
normalizer of algebras and groups
Hi,
I am looking at inclusion of discrete groups $H\subset G$ such that $H$ is abelian and $(hgh^{-1},h\in H)$ is infinite if $g\in G-H$. If you have this, $LH\subset LG$ is a maximal abelian ...
- 363
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 &...
- 5,425
9
votes
4
answers
513
views
Does such a subgroup exist?
I am looking for a certain masa in a $II_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:
$G$ ...
- 363