All Questions
Tagged with hilbert-spaces rt.representation-theory
23 questions
28
votes
2
answers
1k
views
Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 5,898
18
votes
0
answers
373
views
Can Rep(G) tell us whether G is discrete?
Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
- 43.2k
11
votes
2
answers
2k
views
Schur's Lemma for Hilbert spaces
Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...
- 156k
9
votes
2
answers
746
views
Characters of irreducible unitary representations of the Poincaré group
Consider Poincare group $\mathrm{ISO}(1,d-1)$, given by $\mathbb R^{1,d-1}\rtimes SO(1,d-1)$ in signature $(1,d-1)$, for some odd $d \geq 3$.
Denote the universal cover of the component connected to ...
- 533
8
votes
3
answers
1k
views
When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 41.4k
5
votes
1
answer
298
views
Fell's trick for Lie groups
Let $\Gamma$ be a countable discrete group and $\lambda$ the left regular representation of $\Gamma$ on $l^2(\Gamma)$. Let $\rho:\Gamma\rightarrow U(H)$ be a unitary representation of $\Gamma$ on some ...
- 1,903
5
votes
1
answer
495
views
When does an irreducible unitary real representation remain irreducible after complexifying it?
Consider a unitary real representation of a Lie group $G$ over a real Hilbert space $\mathcal{H}_\mathbb{R}$
\begin{equation}
\rho:G\rightarrow U(\mathcal{H}_{\mathbb{R}})
\end{equation}
Taking the ...
- 275
4
votes
2
answers
525
views
When are finite-dimensional representations on Hilbert spaces completely reducible?
Let $G$ be a group and $\pi$ be a finite-dimensional (not necessarily unitary) representation of $G$ on a complex Hilbert space $H$. We shall say that $\pi$ is completely reducible if there exists a ...
- 265
4
votes
1
answer
466
views
Injection between non-isomorphic irreducible Hilbert space reps?
I must be missing something trivial here.
Let $G$ be, say, a reductive Lie group (or more generally any locally compact Hausdorff unimodular topological group). A unitary Hilbert space representation ...
- 41.4k
4
votes
2
answers
313
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
- 692
3
votes
1
answer
176
views
Dense subspace of $\operatorname{Ind}_{H_1 \times H_2}^{G_1 \times G_2} \chi$
Let $H = H_1 \times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \times G_2$, with $H_i \leq G_i$. Let $\chi = \chi_1 \otimes \chi_2$ be a unitary ...
- 6,180
3
votes
0
answers
171
views
Decomposition of $L^2(\Gamma \backslash H)$ into irreducible representations using the spectral theorem
I'm reading the introduction of An Introduction to the Trace Formula by James Arthur and wanted to understand something in the introduction.
Let $H$ be a unimodular locally compact Hausdorff group, ...
- 6,180
3
votes
0
answers
144
views
Deformation and Representations
Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...
2
votes
1
answer
205
views
Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 657
2
votes
0
answers
92
views
Representations of the commutation relations and renormalization
I had a hard time deciding whether this question is more appropriate for physicsSE or MO, so I have cross-listed it for the time being. The physicsSE post can be found here.
In the lecture Is ...
- 721
2
votes
0
answers
220
views
Ultraviolet divergences of entanglement entropy in QFT
I've often read that entanglement entropy in quantum field theory is ill-defined because local algebras are generally of type III, which implies that a trace doesn't exist. For a normal state $\omega_{...
- 424
2
votes
0
answers
147
views
About normal states in abstract von Neumann algebras
In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...
- 424
2
votes
0
answers
107
views
The density of the image of a unitary irrep (a generalization of Burnside's theorem)
I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...
- 161
2
votes
0
answers
222
views
Complete reducibility, in linear algebra and in topology
I thought that this is a simple question and asked it at the Mathematics StackExchange, but I now have to elevate it to MathOverflow.
Consider a representation $A(G)$ of a group $G$ in a vector space $...
- 603
1
vote
0
answers
39
views
Classifying endomorphisms of a direct sum Hilberts pace
Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$...
- 11
0
votes
0
answers
74
views
A question on projective unitary representation of a Lie group
$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
- 287
0
votes
0
answers
114
views
Norm of the Linear Operator
Let $G$ be a compact group, and $\pi : G \rightarrow \mathcal{U}(H)$ be a continuous unitary representation. Let $f \in L^{1}(G)$ be arbitrary.
By Riesz Representation Theorem we can find a bounded ...
- 581
0
votes
0
answers
132
views
Lie algebra action Whittaker model
Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
- 145