3
votes
1answer
32 views

Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
2
votes
0answers
57 views

Is every unitary representation a direct sum of irreducible subprepresentations?

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the ...
4
votes
1answer
31 views

restriction of unitary operator is unitary?

Let $\mathcal{U}: \mathcal{H} \rightarrow \mathcal{H}$ be a unitary operator on a Hilbert space $\mathcal{H}$. If $\mathcal{K}\subset \mathcal{H}$ is a closed subspace such that ...
1
vote
2answers
43 views

Representation of dense Subset

let $\mathcal B \subset \mathcal A$ a dense subset of a C*-algebra $\mathcal A$. I have a representation for $\mathcal B$. Can I then conclude that this is somehow also a representation for ...
1
vote
0answers
32 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
5
votes
1answer
78 views

Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
2
votes
1answer
44 views

Representation in Banach space and norms 'induced' by representation

By $G$ we denote some compact group, $X$ stands for some Banach space. Suppose $\pi\colon G\longrightarrow \mathrm{GL}(X)$ to be some representation in $X$. I'm trying to prove that there is an ...
3
votes
0answers
66 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
4
votes
1answer
59 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
2
votes
0answers
111 views

Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
7
votes
4answers
239 views

What makes irreducible representations nice?

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation. What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations ...
4
votes
1answer
159 views

Cyclic vectors of an irreducible representation of a C*-algebra

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi)$ an irreducible representation of $\mathcal{A}$. I want to prove the statement: all $\xi \in H$ are cyclic or $\pi(\mathcal{A})=\{0\}$ and ...
4
votes
2answers
92 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
3
votes
1answer
99 views

Unitary Equivalence of Two Irreducible $ * $-Representations of a GCR $ C^{*} $-Algebra that Have the Same Kernel.

In general, if two irreducible $ * $-representations of a $ C^{*} $-algebra $ A $ have the same kernel, then we can say that they are approximately unitarily equivalent. When $ A $ is GCR, how can we ...
4
votes
1answer
74 views

irreducible representation of a group

Reduced group $C^\ast$-algebra of group $G$ is defined to be $G^*_{r}(G)=\overline{\lambda(L^1(G))}$ where $\lambda$ is left regular representation. My question is how to get a irreducible ...
1
vote
0answers
71 views

Does a $C^*$ subalgebra of the centralizer of a unitary representation always contain the unit?

I am studying a theorem in Folland's "Course in Abstract Harmonic Analysis" where the following ingredients/assumptions are needed: $G$ a locally compact group, $\pi$ a unitary representation of ...
1
vote
1answer
223 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
6
votes
1answer
122 views

Why need two directions to make $\sim_{\rm wa}$ an equivalence relation?

Let $\pi$ and $\sigma$ be representations of a $C^*$-algebra $\mathcal{A}$. They are weak approximately equivalent ($\pi\mathbin{\sim_{\rm wa}}\sigma$) if there are sequences of unitary operators ...
5
votes
1answer
219 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
4
votes
1answer
151 views

Free groups and Kazhdan's property (T)

Showing non-amenability of a (non-abelian) free group is somewhat easy and one can do this immediately after the definition of amenability. Is there an easy proof of the fact that free groups do not ...
6
votes
1answer
275 views

Topology on the tensor product of two topological vector spaces — how properties does it maintains?

Good morning, this is my first question in this website. If I have two topological vector spaces, say $A$ and $B$, I would like to know 1)how the topology on $A\otimes B$ is canonically defined? ...
4
votes
2answers
329 views

An hermitian operator problem

It is possible to have two hermitian operators $A$ et $B$, with : $B^2 = \mathbb{I}d$ $[A,B] = i * \mathbb{I}d$ where $i$ is the usual (complex) square root of $(-1)$, and $\mathbb{I}d$ is the ...
4
votes
2answers
129 views

Extensions and Kazhdan's Property (T)

Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?