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Sep
10
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
Assume $f$ is a genuine group homomorphism, I.e.,$f(x+y)=f(x)+f(y)$ holds for all $(x,y)\in G\times G$, so in your definition of $F$, $f(x+y)+f(-y)=f(x)$, then $F(x)=f(x)$ for all $x\in G$, so how do you know $F$, which is now $f$ is continuous? To be honest, whenever I left some comments, you kept ignoring what I am talking about, I do not know why.
Sep
10
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
Suppose $f$ is a group homomorphism, then by definition of your $F$, $F(x)=f(x)\forall x\in X$, but we only assume $f$ is measurable, so you can not say that $F$ is continuous.
Sep
10
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
.. more details?I suspect we would run into trouble of some suttle issues that Kleppner faced.
Sep
10
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
Here are a couple of things I want to say. 1, when I see $a.e., x,y\in G$ I would understand it as for some conull subsets $V,W\subset G$, $x\in V, y\in W$ instead of what you mentioned above. 2, suppose some statement P holds for $(x,y)\in Z$ where $Z\subset G\times G$ is conull set. In general, you CAN NOT find conull $V,W\subset G$ such that P holds for $x\in V,y\in W$, since in general $Z$ is complicated, say $Z=G\times G-diagonal$. 3, In your answer $F(x)=f(x) a.e., x\in G$ is true, but I do not see how to show $F$ is continuous, a group homomorphism. If you have time, could you provide..
Sep
9
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
Yes, you can show that $f(x+y)+f(-y)-f(x)=0$ for a.e. $(x,y)\in G\times G$. But the point is that you claim it holds for a.e. $x\in G$ and a.e. $y\in G$, i.e., you claim that you can find $V, W$, both are conull subset in $G$ such that the above equality holds for $x\in V, y\in W$. Could you tell me why? How do you argue that from $Z$ you can get $ V, W$?
Sep
6
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
both are true. Is it related to my above comment?
Sep
6
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
"...iff the measure of the slices of $G\times G\Z$ is zero a.e., by Fubini". I still do not see how to use the assumption that $f(x+y)=f(x)+f(y), \forall (x,y)\in Z" to get $f(x+y)+f(-y)=f(x)$ for a.e. x, a.e. y in G.
Sep
6
comment Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism
$Z\subset G\times G$ is not necessarily of product type.
Aug
12
comment a measurable function on a LCA group coincide with an mulitplicative character almost everywhere
Hi, could you tell me where you have met this fact? I asked an essentially same question recently, and I want to understand this stuff better.
Aug
11
comment Existence of proper I.C.C. subgroup
my fault, I should mention that $G$ is inner amenable group if there exists a finite additive probability measure on the power set of $G$ which is invariant under conjugate action of $G$.(Recall for amenable group, we require that measure is invariant under left action of $G$).
Aug
11
comment induction exercise doubt
Hint: When do induction, you can not change the assumption, in your case, the assumption, $x_1x_2... x_{n+1}=1$ is the assumption, but you could translate your assumption into different form. Try set $x_i'=x_i for 1\leq i\leq n-1, x_n'=x_nx_{n+1}$.
Aug
11
comment What is the closed form for this sequence, powers of $4$?
try oeis.org, but it seems does not work...
Aug
11
comment Proof of Cauchy's Lemma in the case that G is abelian
Do reduction on order of $G$. As you did, take $U$, when $r>0$, focus on $U$, when $r=0$, consider $G/U$.
Aug
11
comment Existence of proper I.C.C. subgroup
Thanks for your quick answer, do you know any I.C.C. non amenable inner amenable groups satisfy this property?
Aug
11
comment Existence of proper I.C.C. subgroup
I do not understand why "g is contained in two separate cyclic subgroups". I think the following argument also works. Clearly, we can pick some $h_i$ with $h_i\not\in \langle g\rangle$ then $\langle h_i, g\rangle=T$ so $T$ is abelian, a contradition.
Aug
11
comment Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.
Why $\mathbb{C}1 \otimes C^*(\pi(c_0(G)), \{\lambda_g : g \in G\}) \cong \mathcal{K}(\ell^2(G))$? And a first feeling is that $\lambda_g$ is not a compact operator.
Aug
11
comment Almost everywhere measurable homomorphism
Could you give any reference for this result?
Jul
27
comment Why is the natural map from maximal to reduced C star algebra surjective?
I see, the second fact you mentioned is also used to show $||.||_{max}$ is the largest norm on algebraic tensor of two C star algebras, thanks a lot!
Jul
26
comment Question on amenable direct summand
Oops, sorry, I should have mentioned $A\subset B\subset N$.
Jul
19
comment find element in relative commutant of a matrix subalgebra
Thanks, I realized that the basic idea is to integral over the unitary group of $M_2(\mathbb{C})$ of $uau^{*}$, matrix units seems to play the role of unitary.