The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

learn more… | top users | synonyms (1)

1
vote
1answer
33 views

A question about state on tensor product of $C^\ast$-algebra

On the page78 of the book C*-algebras and Finite-dimensional Approximations there is a corollary as follows. Corollary 3.4.3. Let $\|\cdot\|_\alpha$ be a $C^\ast$-norm on $A\odot B$ and $\xi$ be a ...
0
votes
1answer
33 views

Is the composite of a state and a $\ast$-homomorphism again a state?

If $\cal{A,B}$ are unital $C^\ast$-algebras and $\varphi:\cal{A\to B}$ is an unital $\ast$-homomorphism. Then it is clear that $\rho\circ\varphi$ is a state on $\cal{A}$ for any state $\rho$ on ...
1
vote
0answers
45 views

Smoothness of distributions

I've reached an impasse in reading some texts on distribution theory, as several of them mention smooth distributions, but none of them actually define what it means. Therefore I'd like to know if ...
0
votes
1answer
51 views

A representation of von Neumann algebra of type I

I am reading a book "C*-algebras and Finite-Dimensional Approximations". There is a quotation below: For infinite-dimensional Hilbert space $H$ and a abelian von Neumann algebra $A$, we can represent ...
2
votes
1answer
66 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...
0
votes
1answer
36 views

A inequality about pointwise absolute value vectors

Let $\Gamma$ be a discrete group and $\xi\in l^{2}(\Gamma)$ be a unit vector. If $|\xi|$ be the pointwise absolute value of $\xi$, then how to verify: ($S$ is a linear bounded operator on ...
1
vote
1answer
23 views

A question about states for nonunital $C^\ast$-algebras

Let $\rho$ be a state on a nonunital $C^\ast$-algebra $\cal{A}$, $a\in\cal{A}$ and $\rho(a^\ast a)\neq0$. Define $\rho_a:b\mapsto\rho(a^\ast ba)/\rho(a^\ast a)$. Then $\rho_a$ is a state on $\cal{A}$. ...
1
vote
0answers
25 views

A certain product of C*-algebras

So, I am looking for some kind of 'product' $\bullet$ on the category of (unital?) $C^*$-algebras satisfying that $M_n(\mathbb{C})\bullet M_m(\mathbb{C}) = M_{m+n}(\mathbb{C})$ where ...
0
votes
1answer
38 views

Is this subalgebra of a semisimple algebra semisimple?

Let $A$ be a semisimple algebra and $e$ be an idempotent in $A$. Then $eAe=\{eae:a\in A\}$ is a subalgebra of $A$ with $e$ as the identity. We want to prove that $eAe$ is also semisimple. That is, if ...
1
vote
1answer
36 views

The integral about probability measures

Definition For a discrete group $\Gamma$, we let Prob$(\Gamma)$ be the space of all probability measures on $\Gamma$: $$Prob(\Gamma)=\{\mu\in l^{1}(\Gamma): ...
5
votes
1answer
98 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
0
votes
1answer
89 views

How do you prove $L^{\infty}$ is a C*-algebra?

If we define on $L^{\infty}$ the essential supremum norm ($\| \|_{\infty}$), then how can I prove this norm is submultiplicative ($\| T_1T_2\|_{\infty}\leq \| T_1\|_{\infty}\|T_2 \|_{\infty}\, \forall ...
1
vote
2answers
47 views

Why does the order on positive elements respect the order on the norm?

The question title doesn't quite convey what I mean, but close enough. I'm struggling with a bit of Davidson's "C* algebras by example", in his proof of Lemma 1.4.7. Our hypotheses: $0 \leq A \leq B$ ...
4
votes
1answer
52 views

Is C(E)a dual of any linear norm space?

Let $E$ is a closed bounded set of $\mathbb{R}$. Is $C(E$ a dual of any linear norm space?
1
vote
1answer
80 views

Universal property about discrete group in C*-algebra

Universal property: Let $u:\Gamma \rightarrow B(H)$ be any unitary representation of $\Gamma$. Then, there is a unique $*-$homomorphism $\pi:C^{*}(\Gamma) \rightarrow B(H)$ such that ...
1
vote
1answer
87 views

Amenable group in C*-algebra

Definition 2.6.1. A group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
0
votes
1answer
71 views

Positive definite function on discrete group in C*-algebra

Recall A function $\phi: \Gamma\rightarrow\mathbb{C}$ is said to be positive definite if the matrix $$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set $F\subset ...
1
vote
1answer
32 views

A simple question about 1-norm

Let $\Gamma$ be a discrete group, if $\mu \in l^{1}(\Gamma)$, then what is the 1-norm of $\mu$, I mean $||\mu||_{1}=?$. As we know, $l^{1}(\Gamma)=\{(\alpha_{x})_{x\in\Gamma}: ...
1
vote
0answers
198 views

Cyclic Properties of the trace in Quantum Field Theory

I'm trying to figure out what's going on in this paper here between lines 10 and 11. But I'll give a brief rundown of what's going on. We are trying to compute the trace of the exponential operator ...
0
votes
1answer
133 views

A question about reduced C*-algebra of discrete group

There is a quotation below: Let $\Gamma$ be a discrete group and $\Lambda\subset \Gamma$ be a subgroup. The right cosets give a direct sum decomposition $$l^{2}(\Gamma)\cong\bigoplus l^{2}(\Lambda ...
0
votes
1answer
51 views

An easy (I guess) question about vector state in C*-algebra

I meet with some problems when I read a book about C*-algebra. Definition 2.5.10. Let $\phi:\Gamma \rightarrow \mathbb{C}$ be a function ($\Gamma$ is a discrete group here). We define a corresponding ...
2
votes
2answers
122 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
19 views

A simple question on infinite dimensional von Neumann algebra

Recall a projection $p\in N$ is called abelian if $pNp$ is an abelian algebra. If $N$ is a von Neumann algebra without abelian projections, then can we conclude that $N$ must be infinite dimensional? ...
1
vote
1answer
43 views

About Fourier transform

The reduced C*-algebra of $\Gamma$, denoted $C^{*}_{\lambda}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm $$\|x\|_{r}=\|\lambda(x)\|_{\mathbb{B}(l^{2}(\Gamma))},$$ The ...
1
vote
1answer
39 views

Squares of C*-algebras

I'm reading a paper where it is claimed that every C*-algebra $A$ satisfies $A^2 = A$, "for example, using Cohen's 1959 factorization theorem". However, I don't see how to apply Cohen's factorization ...
1
vote
1answer
35 views

A question about the positive definite function

Definition 2.5.6. A function $\phi:\Gamma \rightarrow \mathbb{C}$ is said to be positive definite if the matrix$$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set ...
2
votes
2answers
221 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
0
votes
1answer
49 views

A question about compact operator

For a discrete group $\Gamma$, $T\in \mathbb{B}(l^{2}(\Gamma))$ is constant down the diagonals-meaning that for every $s, t, x, y\in \Gamma$ such that $ts^{-1}=yx^{-1}$, we have $\langle T\delta_{s}, ...
0
votes
1answer
95 views

A question about full group C*-algebra

There is a quotation below: $\qquad$The $full~group$ C*-algebra of $\Gamma$, denoted $C^{*}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm ...
0
votes
1answer
39 views

The one to one map between two representations

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular ...
1
vote
1answer
24 views

Showing the $C^*$ identity

I'm working through a proof in Dixmier's book on $C^*$-algebras and I'm stuck on part of a proof. I'm given a Banach algebra $\mathcal{A}$ which has norm $\lVert\cdot\rVert$ and a semi-norm ...
0
votes
1answer
43 views

A question on left regular representation of a discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}~$ for all ...
3
votes
1answer
81 views

Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
2
votes
1answer
143 views

Full (or universal) group C*-algebra of discrete group $\Gamma$

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums ...
0
votes
2answers
51 views

A question on discrete group

There is a quotation below: For a discrete group $\Gamma$ , $f\in l^{\infty}(\Gamma)$ and $s, t\in \Gamma$. We let $s.f \in l^{\infty}(\Gamma)$ be the function $s.f(t)=f(s^{-1}t)$. My question is ...
0
votes
1answer
129 views

Support and range projections in von Neumann algebra

There is a quotation below: Let $M$ be a von Neumann algebra, take a noncentral projection $p\in M$ and find some $m\in M$ such that $pm(1-p)\neq0$. The partial isometry in the polar decomposition of ...
1
vote
2answers
60 views

A question about discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, ...
0
votes
1answer
39 views

How do I compute the specific map between two isomorphic finite C* algebras?

Starting with a finite C* algebra $\mathcal{A} \subset M_{n}\left({\mathbb C}\right)$ (complex $n\times n$ matrices), $\mathcal{A}$ is known to be isomorphic to a canonical algebra of the form ...
2
votes
3answers
66 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
0
votes
1answer
38 views

Question about special $C^*$-algebra

i have a question about a $C^*$ algebra $A$ namely $M_2(\mathbb{C})$. I want to prove that every state $\alpha$ of $M_2(\mathbb{C})$ (thus a positive linear functional with norm $1$) is of the form ...
1
vote
1answer
30 views

Strong operator sum of corner projections is a normal map

Suppose that we are given a Hilbert space $H$ with an orthogonal basis $(e_i)_{i\in I}$ and let $P_i$ denote the projection of $H$ onto $\mathbb{C}e_i$. Then we can consider the map ...
0
votes
1answer
22 views

Images of unitaries

Let $n\geqslant 0$. Suppose that $U$ is a unitary matrix in $M_n$ and there are two unital ${}^\ast$-homomorhpisms $\pi_1\colon M_n\to A, \pi_2\colon M_n \to B$, where $A,B$ are C*-algebras such that ...
3
votes
1answer
29 views

Unit in the image of a cp map

This is another question which looks non-trivial to me. Suppose that we have a completely positive map $f\colon M_n \to M_m$ such that $f(a) = I_m$, the identity matrix on $M_m$. Is there a positive ...
0
votes
1answer
49 views

The direct sum of two nuclear C*-algebra

Recall: Definition 2.1.2 If $A$ is a C*-algebra and $N$ is a von Neumann algebra, a map $\theta:A \rightarrow N$ is called weakly nuclear if there exist c.c.p. maps $\phi_{n}: A\rightarrow ...
1
vote
1answer
73 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
2
votes
1answer
42 views

Are contractive completely positive maps trace decreasing?

Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is ...
0
votes
1answer
42 views

An exercise about nuclear C*-algebra

Definition 2.3.1. A C*-algebra $A$ is nuclear if the identity map id$_{A}:A \rightarrow A$ is nuclear. Exercise 2.3.7. If for each finite set $F\subset A$ and $\epsilon>0$ one can find a nuclear ...
1
vote
1answer
74 views

A question about nuclear C*-algebra

Definition 2.3.1. A C*-algebra $A$ is nuclear if the identity map $id_{A}: A\rightarrow A$ is nuclear. Definition 2.3.2. A C*-algebra $A$ is exact if there exists a faithful representation $\pi:A ...
3
votes
0answers
30 views

A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a ...
0
votes
1answer
42 views

Proof trick for C*(F$_2$) has finite dimension faithful representations

The image is from the C*-algebras by Example by Kenneth R. Davidson, here F$_2$ is the free product of Z$_2$ and Z$_3$. I have two questions: 1) Why do we have the matrices U$_n$ and V$_n$, I guess ...