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
2answers
63 views

Unitarily equivalent $C^*$-algebra representations

the situation i want to talk about is the following: $(H_1,\varphi_1),(H_2,\varphi_2)$ irreducible representation of a $C^*$-algebra $A$. A bounded operator $T:H_1\rightarrow H_2$ such that ...
1
vote
1answer
70 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
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
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
237 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
158 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 ...
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 ...
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 ...
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 ...
2
votes
1answer
112 views

Some operators on $\ell^2$ of a discrete group: are they a von Neumann algebra?

Caution: Written while trying to turn some vague confusion I was having into precise questions. A bit long and rambling. An appropriate way to answer may just be to point me toward references. Let ...
2
votes
0answers
173 views

clarification on the definition of a group C*-algebra

I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the ...