All Questions
Tagged with geometric-group-theory oa.operator-algebras
15 questions
12
votes
0
answers
373
views
Does Thompson's group $V$ have property AP?
Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
11
votes
1
answer
381
views
Are groups with the Haagerup property hyperlinear?
In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
- 1,027
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
7
votes
2
answers
871
views
Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
- 175
7
votes
0
answers
193
views
Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements
Consider the free product of $\mathbb{Z}/2$ with itself with generators
$$
\mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle
$$
and regard its group $C^*$-algebra
$$
C^*(\mathbb{Z}/2*\mathbb{...
- 598
5
votes
1
answer
237
views
Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2
In this recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C*-algebras. The answer, provided by Caleb Eckhardt in a comment, was ...
- 2,263
5
votes
0
answers
117
views
Finitely presentable group with purely infinite full group $C^*$-algebra?
Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite,
resp. is it known to be impossible?
- 598
4
votes
1
answer
263
views
Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations
A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...
- 1,090
4
votes
0
answers
172
views
$S^{3}$-valued harmonic analysis
Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid \phi(xy)=\phi(...
- 356
4
votes
0
answers
268
views
Can one pose a Toeplitz index problem associated to a discrete group?
Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking.
Let's start with the classical case of the Toeplitz index problem on the circle....
- 29.2k
3
votes
1
answer
230
views
An inequality for Fuchsian groups?
Let $G$ be a finitely generated Fuchsian group.
(i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$).
Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ?
Here, $\beta_{2}^1(G)$ stands for the ...
- 11.3k
3
votes
1
answer
393
views
Kaplansky Idempotent conjecture and Extension theory
We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
- 356
2
votes
0
answers
92
views
Property A, Higson-Roe condition and its applications
Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition:
Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...
- 21
1
vote
0
answers
156
views
Property T Versus Property A
Kazhdan Property T for a locally compact group is defined in terms of the unitary dual. Namely, a group has property $T$ if the identity is an isolated point in the space of ...
- 2,630
1
vote
0
answers
147
views
Is there a non-integer in the dimension spectrum for the Heisenberg group?
Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group.
Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...
