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A finite abelian group has a faithful irreducible representation if and only if it is cyclic. The case of finite groups was solved by Gaschütz in W. Gaschütz, Endliche Gruppen mit treuen absolut-irred …

The situation in infinite dimensions is different for Kazhdan groups. If $\Gamma$ contains a non-abelian free group, then the left-regular representation $\lambda \colon \Gamma \to U(\ell^2 \Gamma)$ …

The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor $$Rep_G …

The natural map $$GL_n \left(\lim_{\leftarrow} (A/{I_p}) \right) \to \lim_{\leftarrow} \ GL_n(A/{I_p})$$ is always an isomorphism. Just note that $$\lim_{\leftarrow} \ GL_n(A/{I_p}) \subset \prod_{p …

This answer shows that one cannot find $(G,\rho)$ as required if $G$ is supposed to be group of Lie type defined over a large field. Let $G$ be a group of Lie type defined over a field with $q$ eleme …

The question depends very much on the regularity that you demand. You have to decide before asking the question which operators are supposed to be self-adjoint or merely symmetric as unbounded operato …

If you consider pairs of unitaries instead, and the group, that you get out of a similar construction, the answer is negative. That is an analogous question but in a slightly different context. The ne …

The answer is yes, this always holds. Note that $$\dim(im(f)) \cdot \|f\|^2 \cdot | {\rm supp}(f)| \geq \tau(f^*f) \cdot |{\rm supp}(f)| \geq |G| \cdot \|f\|^2_1.$$ Here, $\tau \colon \mathbb C[G …

I do not think that there is a complete answer to this question. However, one can give some necessary conditions (which show that any answer must be complicated). One can show that a triple $(x,y,z) …

If $G$ is amenable, then every weakly mixing representation $\pi$ has almost invariant finite-dimensional subspaces. This means that $\pi \otimes \bar \pi$ has almost invariant vectors. Results like …

Some of the crucial ideas and results on quasi-abelian categories in the context of topological vector spaces are already contained in the work of Lucien Waelbroeck (see here) which predates the work …

Setting $a_0=A^7, b_0=B^7$ and then $a_{n+1}=[b_n^{-1},a_n], b_{n+1}=[a_n,b_n]$ seems very efficient. The length grows like $C \alpha^n$ with $\alpha= \frac{3 + \sqrt{17}}2$ and the operator norm dist …

The answer is no. Consider the Toeplitz algebra $\mathcal T$ with its canonical representation on $\ell^2 \mathbb N$, which is generated as a $C^\star$-algebra by the shift $S(e_n)=e_{n+1}$. It is we …

Daniel Quillen has answered this in Quillen, Daniel G., On the associated graded ring of a group ring. J. Algebra 10 1968 411–418. From the Mathematical Reviews by J. Knopfmacher: "Let $KG$ denot …

If $n:=[G:H]$, then $\mathbb C[G] \subset M_n \mathbb C[H]$, where $g \in G$ maps to a permutation matrix decorated with elements from $H$ and the embedding depends essentially only on a choice of a t …