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Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ t …

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the L …

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous $$ x_i\to x\quad\Longrightarrow\quad ||\varphi( …

I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here. Hewitt and Ross define trigonometric polynomial on a locally compact …

I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my …

Let $G$ be a finite group, $N$ its normal subgroup, and $\pi:N\to{\mathcal B}(X)$ a unitary representation of $N$ in a Hilbert space $X$. Consider the induced representation $\pi':G\to{\mathcal B}(L_2 …

Recently I found in the web a discussion on the following question: For which locally compact group G its norm-contunuous unitary representations separate points of G? (A unitary representation $\p …

If a topological vector space $X$ is not locally convex, then it usually has not non-zero linear continuous functionals, and this means that there are no non-trivial continuous characters on $X$. For …

Let $G$ be a Lie group which is a finite extension of an open normal subgroup $N$: $$ 1\to N\to G\to F\to 1 $$ (so $N$ and $G$ are Lie, and $F$ is finite; but I think, this is not very important, we c …

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without success …

There is a direct way to define group algebras for different classes of groups (locally compact groups, Lie groups, algebraic groups, etc.). For an (affine) algebraic group $G$ one should take the a …

A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short w …

I am sorry, I have realized that the answer is "yes", and this is simple. The proof is the following. Suppose this is not true. Then we can find a locally compact group $G$ which is not locally Euclid …

I asked this 11 days ago at MSE, but there was no answer, I hope people here could help. Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is sa …

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) …