0
votes
0answers
26 views

Scalar Products on the Rational Function Field

Let $\mathbb{R}(t)$ be the rational function fields over $\mathbb{R}$. Are there scalar products $\langle -,- \rangle$ on $\mathbb{R}(t)$ such that multiplication with $t$ is selfadjoint, i.e. ...
2
votes
1answer
32 views

The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing ...
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
67 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 ...
0
votes
0answers
47 views

self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
6
votes
1answer
137 views

Question on irreducible representation of a Banach algebra

Let $\mathcal A$ be a Banach algebra over $\mathbb{C}$, $\mathcal X$ a irreducible left $\mathcal A$-module. If $x,y \in \mathcal X$ are linearly independent, there exists an element $a\in\mathcal A$ ...
2
votes
0answers
107 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 ...
4
votes
1answer
157 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 ...
1
vote
1answer
219 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 $X$. ...
2
votes
2answers
154 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
1
vote
1answer
45 views

Where can I find an English version of this paper by Gel'fand and Raikov?

It is titled 'Irreducible unitary representations of locally bicompact groups' and the original version is in Russian. Google scholar shows it has been translated into English and once pubulished in ...