11
votes
0answers
336 views

Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...
4
votes
0answers
81 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
0
votes
2answers
188 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 $II_1$ ...
2
votes
1answer
334 views

Is every distribution a linear combination of Dirac deltas?

My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution ...
9
votes
3answers
567 views

measure with given push-forwards

Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps ...
10
votes
0answers
127 views

Star-shaped Folner sequence

Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that ...
4
votes
0answers
266 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 ...
4
votes
2answers
342 views

Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic?

Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is: Are these groups ...
5
votes
0answers
213 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 ...
0
votes
1answer
170 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 ...
4
votes
0answers
249 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 ...
0
votes
1answer
243 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 ...
15
votes
2answers
1k views

What is a Gaussian measure?

Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions. Is there a direct ...
7
votes
2answers
459 views

Baum-Connes-like “conjecture” for $l^p$-spaces

Let $G$ be a (discrete) group. For the Baum-Connes conjecture, one looks at the reduced group $C^{\star}$-algebra: Look at the Hilbert space $l^2(G)$ and the representation of $G$ on this Hilbert ...
4
votes
1answer
230 views

left- and right- Folner sets

Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m ...
5
votes
0answers
288 views

The kernel of all invariant means

Let $G$ be discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional on $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g ...
0
votes
0answers
86 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 ...
18
votes
1answer
868 views

Existance of certain almost invariant functions related to amenability and piece-wise transformations

We would like very much to know the answer to the following question: Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
4
votes
3answers
362 views

What classes of functions are closed under all rescalings?

Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables ...
1
vote
4answers
316 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 ...
9
votes
0answers
331 views

Lacunary hyperbolic groups and weak amenability

In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
2
votes
1answer
484 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 ...
5
votes
1answer
488 views

Cosets of groups of functions

Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$. The set $\mathcal ...
9
votes
2answers
856 views

Property (T) for pseudogroups

Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). ...
5
votes
3answers
849 views

Invariant means on the integers

Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density ...
4
votes
1answer
574 views

Subgroups of U(M_n)

can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite sets by a finite subgroup? i.e. is the following True or false? Let $n, d$ be positive integers ...
5
votes
0answers
275 views

Is there an idempotent measure on the free LD system?

This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: ...
13
votes
1answer
671 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
13
votes
1answer
353 views

Self map of unitary group

Let $H$ be a Hilbert space and let $u_1 \in U(H)$ be a unitary operator on $H$. Consider the self-map $w: U(H) \to U(H)$ which is given by $$w(v) := v^2 u_1 v^{-1}.$$ Since $U(H)$ is connected, there ...
4
votes
2answers
330 views

Is every bounded representation of Z unitarisable when all sets are measurable?

For the purpuse of this question, a group is amenable iff there exists a Foelner sequence. Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...
16
votes
13answers
3k views

Is there a “crash-course” book on Abelian varieties (e.g., an introduction for physicists)?

Hello, In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...
1
vote
2answers
416 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 ...
4
votes
1answer
205 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 ...
2
votes
2answers
402 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 ...
6
votes
1answer
359 views

Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?

Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}_1$ and let $\mathbb{F}_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}_n$ on $\mathcal{R}$. Does the von ...
10
votes
2answers
515 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...