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9 votes
1 answer
435 views

Questions on the group $\mathrm{GL}(H)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$. Question 1. I've ...
3 votes
0 answers
205 views

Status of RFD groups and $C^*$-algebras

Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
8 votes
0 answers
251 views

When does a semisimple $\mathbb{C}$-algebra come from a group?

Let $\mathcal{A}$ be a semisimple $\mathbb{C}$-algebra. By the Artin-Wedderburn theorem, it is isomorphic to a direct product of matrix algebras: $$ \mathcal{A} = \prod_{i=1}^m M_{n_i}(\mathbb{C})$$ ...
6 votes
2 answers
875 views

Is the set of all ICC amenable groups countable?

Is the set of all ICC amenable groups countable? If "yes", then in general, the classes of all countable ICC groups that give rise to the same von Neumann algebra (factor) -- are these ...
0 votes
1 answer
326 views

Group algebras and group automorphisms

Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and ...
1 vote
0 answers
229 views

Tensor product decomposition of commuting representations

If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
9 votes
0 answers
290 views

A robust version of Schur's lemma?

Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this: Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
9 votes
1 answer
521 views

Which group algebras in analysis are "true group algebras"?

Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that $$ \pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...
11 votes
2 answers
537 views

Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which 1- is finitely generated by $S$, 2- does not have property (T), 3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...
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 ...
3 votes
1 answer
506 views

Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the "other" Schur multipliers of a group?

The name for the the following 2 mathematical objects: $$H_2(G,\mathbb{Z})$$ and $$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times G\longrightarrow\...
5 votes
0 answers
321 views

Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
12 votes
3 answers
1k views

Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors: Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset R^{H_{1}}...
18 votes
2 answers
1k views

Regarding Cayley Graphs of Property (T) Groups

A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...
2 votes
1 answer
298 views

An upper bound for the maximal subgroups at fixed index?

Let us call a subgroup an injective homomorphism between groups. I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone. A subgroup $H \subset G$ is ...
9 votes
2 answers
614 views

Dimensions of unitary representations of group extensions

Is the property of having a bound on the dimensions of irreducible representations preserved by an extension? For example $G_1=\mathbb{Z}$ and $G_2=\mathbb{Z}/2\mathbb{Z}$ are discrete abelian ...