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
2answers
65 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 ...
0
votes
1answer
14 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
37 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
29 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
23 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
106 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
39 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
49 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
29 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
21 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
31 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
39 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 ...
1
vote
1answer
51 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
46 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
64 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
43 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
29 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
55 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
35 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
20 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
21 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 ...
2
votes
1answer
28 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
39 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
63 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.
0
votes
0answers
17 views

Tensor product decomposition for algebras with non trivial center

I have a question regarding operator algebras with non-trivial center. This is with regard to defining entanglement entropy in gauge theories. Suppose there exists an algebra of operators associated ...
2
votes
1answer
35 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
29 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
51 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 ...
1
vote
0answers
22 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 ...
1
vote
1answer
27 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 ...
0
votes
2answers
41 views

A proof of a basic conclusion in operator algebra

There is a quotation below: (in a book named "C*-algebras Finite-Dimensional Approximations") Lemma 2.3.4. Let $A$ be a Banach space, $\mathbb{B}(A)$ be the space of all bounded linear maps from $A$ ...
1
vote
1answer
49 views

Commutator proof

I have a proof in my book I dont fully understand. The author is proving that if $[A,B]=1$ then $[A,B^n]=nB^{n-1}$. The proof is really short, it is only one line of equations, but I dont understand ...
0
votes
1answer
30 views

A question about hereditary C*-subalgebra

Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at infinity My question is : If $P\in M_{n}(C_{0}(X))$ is a projection, then ...
3
votes
1answer
48 views

A isomorphism between C*-algebras

Let $A$ be a C*-algebra and $J\triangleleft A$ be an ideal, then $A^{**}\cong J^{**}\oplus(A/J)^{**}$ ? Why?
0
votes
1answer
24 views

A question about ultraweakly dense

Let $A$ be a c*-algebra, then the positive elements in $M_{n}(A)$ are ultraweakly dense in the positive part of $M_{n}(A^{**})$. I do not know how to prove this conclusion. Could someone show me more ...
1
vote
1answer
48 views

Can anyone give an example of two stably equivalent projections that are not Murray Von Neumann equivalent?

Two projections $P$, $Q$ are MvN equivalent in $C^*$-algebra $A$ when there is an element $u\in A$ such that $uu^*=P$ and $u^*u=Q$, and two projections $P$, $Q$ are stably equivalent if $P\oplus ...
0
votes
1answer
32 views

A simple question on predual in C*-algebra

Let $A$ be a C*-algebra, then $A^{*}=(A^{**})_{*}$? Here, $(.)_{*}$ denotes the predual of $(.)$.
0
votes
1answer
50 views

A question about utilizing Hahn-Banach theorem

There is a quotation below: Let $A$ be a Banach space, $\mathbb{B}(A)$ be the space of all bounded linear maps from $A$ to $A$ and $C\subset \mathbb{B}(A)$ be any convex set. If a net ...
2
votes
0answers
24 views

Space of operators on function

Consider the following space of operators on function of $n$-variables $A= Span \{x_ix_j\ , x_i \frac{\partial}{\partial x_j} , \frac{\partial^2}{\partial x_i \partial x_j} , i,j=1,2,\cdots,n\}$. ...
0
votes
1answer
32 views

Completely positive maps between matrix algebras

Let $n<m$ be natural numbers and consider the C*-algebras $M_n$ and $M_m$ of matrices. Suppose that $f\colon M_n\to M_m$ is a a completely positive (linear) map. Is it true that ...
0
votes
1answer
25 views

Interchange exponential of operators in quantum mechanics

What is the formula for interchanging products of exponential operators in quantum mechanics., i.e. I want to write $e^Ae^B = e^{B+...}e^A$
1
vote
2answers
55 views

An approximation question on projections

Suppose $\{p_i\}_{i=1}^{m}$ are projections in the d by d matrix algebra $A$ over the complex numbers and satisfy the following condition: $||Id-\sum_{i=1}^m{p_i}||_2<c$, $||p_ip_j||_2<c, ...
1
vote
1answer
31 views

An easy question about contractive completely positive map

Recall that a map $\phi: M\rightarrow N$ of von Neumann algetras is normal if $$\phi(sup x_{i})=sup\phi(x_{i})$$ for all norm bounded, monotone increasing nets of self-adjoint elements ...
1
vote
1answer
32 views

An exercise about the definition of nuclear maps

Definition 2.1.1 Let $A, B$ be the C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
0
votes
1answer
25 views

A question on extension of contractive completely positive map

Assume $A$ is a nonunital C*-algebra, $B$ is a unital C*-algebra and $\phi:A \rightarrow B$ is a contractive completely map. Then $\phi$ can extend to a unital completely positive map $\bar{\phi}: ...
3
votes
2answers
49 views

Why $ \|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ in the definition of $C^*$ algebra?

I read the definition of $C^*$ algebra in Wikipedia where it says $\|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ but I do not know why. Can you show me how to derive $\|xx^*\| = ...
0
votes
1answer
32 views

Double adjoint map in C*-algebra

There is a quotation below: Assume $A$ is nonunital C*-algebra and $B$ is unital C*-algebra and $\phi: A\rightarrow B$ is a contractive completely positive map. Consider the double adjoint map ...
2
votes
1answer
25 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
2
votes
1answer
28 views

The comprehension of a paragraph about point-ultraweak convergence

There is a quotation below (in the book "C*-algebras and Finite-Dimensional Approximations") Remark 2.1.3. It follows from Sakai's predual uniqueness theorem that when checking point-ultra weak ...
0
votes
1answer
23 views

A question about completely positive map

Let $A$ be a unital C*-algebra and $\phi: A\rightarrow M_{n}(\mathbb{C})$ be a completely positive map. If $P$ denotes the projection onto the kernel of $\phi(1_{A})$ and $P^{\perp}=1-P$ is the ...