All Questions
Tagged with fa.functional-analysis gr.group-theory
102 questions
4
votes
1
answer
212
views
Kernels of representations of $C^*(G)$
Let G be a discrete group. I am interested in the following: let $\pi$ and $\rho$ be two representations of $G$. Denote by $C^*Ker\pi$ and $C^* Ker \rho$ the kernels of the corresponding ...
- 631
4
votes
2
answers
452
views
Is every bounded representation of Z unitarisable when all sets are measurable?
For the purpose of this question, a group is amenable iff there exists a Følner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...
- 3,897
4
votes
1
answer
204
views
Making Hermitian matrices almost commute
Consider two Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$. I'm interested in finding another Hermitian matrix $A'$ that is close to $A$ and almost commutes with $B$. More precisely, I'd like ...
- 209
4
votes
1
answer
221
views
existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 1,222
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
4
votes
0
answers
195
views
Bounded cohomology and unitary representations
On page 9 of Nicolas Monod's very nice ICM report "An invitation to bounded cohomology" (https://egg.epfl.ch/~nmonod/articles/icm.pdf), he mentions that bounded cohomology may be related to ...
4
votes
0
answers
114
views
Coming up with a represenation for sum of functions in the Fourier algebra
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com.
Let $G$ be a discrete group.
Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 161
4
votes
0
answers
297
views
Which orbits of a separable representation of the infinite unitary group are closed?
Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...
- 965
3
votes
1
answer
361
views
Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
- 175
3
votes
1
answer
213
views
About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$
Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...
- 185
3
votes
1
answer
243
views
Can a non-free Whitehead group embed as a discrete subgroup of a normed space?
Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
- 11.3k
3
votes
1
answer
298
views
Averaging measurable functions over amenable group actions
Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
- 10.4k
3
votes
0
answers
115
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
3
votes
0
answers
93
views
Connection between Schur multipliers in representation theory and functional analysis? [duplicate]
I was wondering if there is any connection between two things called Schur multipliers or is it just a coincidence? Namely, in representation/group theory the Schur multiplier of a group $G$ is its ...
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 ...
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
3
votes
0
answers
237
views
Orthogonality relations for unitary representations of infinite (finitely generated) groups
Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
- 333
3
votes
0
answers
301
views
What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
user23860
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
2
votes
1
answer
246
views
Does this sequence contain a nonnegative number?
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
2
answers
178
views
Point spectrum of a positive invertible operator
Let $G$ be a l.c. group and $f$ belong to $C_c(G)$, the space of continuous functions with compact support. Define an operator$T_f$ on $L^2(G)$ by $T_f(g)=f*g$ (the convolution product). If $T_f$ is ...
- 2,118
2
votes
1
answer
172
views
Ideals of $L^1(G)$ and normal subgroups of $G$
Let $G$ be a locally compact group. Is there any correspondence between closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$? (Especially, is there any correspondence between finite ...
- 565
2
votes
4
answers
411
views
A Fractional Linear Transformation Class Property
Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and $f'(x)>0$...
- 81
2
votes
1
answer
2k
views
Invariant functionals on C(R) and amenable groups
Since there seems to be no progress in this interesting question, I took the liberty to reformulate it in a way, that is easier to understand. Moreover, my answer shows that the question is related to ...
- 385
2
votes
1
answer
99
views
Definite negative functions and length functions
$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups.
Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...
- 21
2
votes
1
answer
156
views
Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)
Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$.
Since all the $...
- 181
2
votes
0
answers
102
views
Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
- 301
2
votes
0
answers
78
views
Delta distribution for compact groups and its derivatives
Let $G$ be a compact group (e.g. $SU(2)$) and $\rho: G \mapsto GL(n)$ a representation of it. Then we can define the delta function
$\delta(g-1)=\sum_{l}\chi_l(g)\chi_l(1) = \sum_l\dim_l(G)\chi_l(g)$
...
- 131
2
votes
0
answers
115
views
Proof that any hyperbolic group has Rapid Decay property
A classic result that states that any hyperbolic group in the sense of Gromov has Rapid Decay property in the sense of Jolisaint. But the original proof of that fact is contained in an old Ph.D. ...
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
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
2
votes
0
answers
224
views
Inner amenability
There is a well-known result of Rosenblatt that if $G$ is non-amenable discrete group acting on the space $X$ and a stabilizer of each point $x\in X$ is amenable then the action of $G$ on itself is ...
- 631
1
vote
1
answer
231
views
Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
1
vote
1
answer
387
views
All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k
The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \...
- 10.5k
1
vote
1
answer
272
views
Is it possible to extend this homomorphism?
Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it ...
- 2,118
1
vote
2
answers
289
views
Any analysis on phase of eigenvalue of unitary matrix?
I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
- 11
1
vote
1
answer
423
views
Quaternion representation and Haar measure of $SU(3)$ [closed]
Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure?
Also, is $SU(2)$ really simplified in the quaternion base?
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
1
vote
1
answer
133
views
Realizing certain affine functions on Choquet simplices on dimension groups
This is a question that is a bit outside my usual mathematical comfort zone, but I feel like an expert might know the answer.
Recall that a dimension group is an ordered abelian group $G$ with ...
- 1,023
1
vote
0
answers
79
views
Groups without "almost equivariant" coarse embeddings
Let $X$ be a set. We say that $\psi:X\times X\to[0,\infty)$ is a CND (conditionally negative definite) kernel if there is a Hilbert space $\mathcal{H}$ and a map $f:X\to\mathcal{H}$ such that
\begin{...
- 81
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
0
answers
203
views
How generic are Cayley graphs of non-Abelian groups with logarithmic girth?
Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...
- 1,893
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
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 $II_1$ ...
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
0
votes
1
answer
187
views
Harmonicity of the Martin kernels
Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum_{n\geq 0}\mu^{*n}(x^{-1}y)$, where $\mu^{*n}$ is the $n$th ...
- 2,090
0
votes
1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
0
votes
1
answer
328
views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407