1
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1answer
25 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 ...
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1answer
43 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 ...
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votes
1answer
32 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 ...
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1answer
29 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}: ...
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votes
1answer
36 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 ...
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votes
1answer
28 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 ...
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votes
1answer
9 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? ...
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1answer
32 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 ...
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1answer
18 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 ...
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1answer
34 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}, ...
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votes
1answer
27 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 ...
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votes
1answer
28 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 ...
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votes
1answer
23 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 ...
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1answer
31 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 ...
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votes
2answers
42 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 ...
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votes
1answer
42 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 ...
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2answers
38 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, ...
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1answer
15 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 ...
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1answer
29 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 ...
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1answer
25 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 ...
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1answer
39 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 ...
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votes
2answers
34 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 $A$ ...
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1answer
20 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
37 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?
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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 ...
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1answer
35 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 ...
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votes
1answer
25 views

A simple question on predual in C*-algebra

Let $A$ be a C*-algebra, then $A^{*}=(A^{**})_{*}$? Here, $(.)_{*}$ denotes the predual of $(.)$.
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1answer
26 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 ...
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1answer
22 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 ...
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votes
1answer
16 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}: ...
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votes
1answer
24 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
20 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 ...
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votes
1answer
20 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 ...
1
vote
1answer
39 views

The matrix in C*-algebra

Let $A$ be a C*-algebra and $A^{**}$ be the double adjoint of $A$. Can we conclude $M_{n}(A^{**})\cong (M_{n}(A))^{**}$?
2
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1answer
35 views

A question on double dual of C*-algebra

Let $A, B$ be the C*-algebra. Assume $A$ is nonunital, $B$ is unital and $\phi: A \rightarrow B$ is a contractive completely positive map. Then we consider the double adjoint map $\phi^{**}: ...
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vote
1answer
30 views

An exercise about nuclear map Von Neumann algebra

There is a quotation below: Let $M\subset B(H)$ be a von Neumann algebra and $\{P_{i}\}_{i\in L}$ be a net of finite-rank projections which increases to the identity (in the strong operator ...
2
votes
1answer
45 views

An exercise on nuclear maps in C*-algebra

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
14 views

The convergent in the point-ultraweak topology

Let $A$ be a C*-algebra , $B(H)$ be the bounded linear operator on Hilbert space $H$ and $P_{i}\in B(H)$ be an increasing net of finite-rank projections which converge to the identity in the strong ...
0
votes
1answer
16 views

Normal completely positive map in C*-algebra

Let $A$ be a C*-algebra, for a linear map $\phi: A\rightarrow M_{n}(\mathbb{C})$, we define a linear functional $\bar{\phi}$ on $M_{n}(A)$ by ...
2
votes
0answers
57 views

A question about positive normal linear functionals in C*-algebra

There is a question below: Let $\phi :B(H) \rightarrow M_{n}(C)$ be a contractive completely positive map. Since $\phi$ is a contractive completely positive map, the corresponding functional ...
1
vote
1answer
35 views

Approximately unitarily equivalent in C*-algebra

There is a quotation below: Definition 1.7.2. Two maps $\pi: A\rightarrow B(H)$ and $\sigma: A\rightarrow B(K)$ are called approximately unitarily equivalent if there is a sequence of unitary ...
4
votes
1answer
52 views

A question about essential representation in C*-algebra

There is a quotation of a book "C*-algebras Finite-Dimensional Approximations" below: Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
2
votes
1answer
67 views

A question on simple and unital $C^\star$-algebra

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations" Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
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votes
1answer
45 views

A proposition about Voiculescu's Theorem in C*-algebra

It is the quotation below: Exploiting the duality between completely positive map $A \rightarrow M_{n}(C)$ and states on $M_{n}(A)$, it is not too hard to deduce the next result from Glimm's lemma. ...
2
votes
1answer
83 views

Arveson's Extension Theorem in C*-algebra

I am reading a book C*-algebra and finite-Dimensional Approximations. There are two conclusions (in the book) below. Corollary 1.5.16. Let $E\subset A$ be an operator subsystem and $\phi: E ...
2
votes
1answer
22 views

Reference for finite Von Neumann algebra

I'm looking for a reference where I can read about very basic properties and examples of finite VN-algebras. I will be happy with one that defines the concept in an "easy" language (for a second year ...
1
vote
1answer
45 views

Completely positive map on C*-algebra

There is a quotation in a book about C*-algebra. A positive linear functional $f$ on an operator system $E$ is completely positive map. Indeed, for $\xi=(\xi_{1}, \xi_{2},\ldots,\xi_{n})\in ...
2
votes
1answer
25 views

A confusion on Stinespring dilation

There is a quotation of Stinespring dilation in a book about C*-algebra. (Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there ...
0
votes
1answer
14 views

A proposition about minimal Stinespring dilation in C*-algebra

Proposition 1. Let $(\pi, \widehat{H}, V)$ be the minimal Stinespring dilation of a contractive completely positive map $\phi: A \rightarrow B$. Then, there exists a *-homomorphism $$\rho: ...
1
vote
2answers
31 views

A proposition about cyclic representation in C*-algebra

Let $A$ be a C*-algebra, if for an arbitrary cyclic representation $\pi: A \rightarrow B(H)$, we have $\pi(a) \geq 0$, $a\in A$, then can we conclude that $a \geq 0$?