4
votes
0answers
12 views

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
0
votes
1answer
35 views

Calculate C*-subalgebra of $A$,$A^*$ and $\mathbb 1$. A Matrix.

I've given a matrix $A=\left( \begin{array}{ccc} 1-3 \cos (2 \lambda ) & 3 i \sin (2 \lambda ) & 2 i \sin (\lambda ) \\ -3 i \sin (2 \lambda ) & 3 \cos (2 \lambda )+1 & 2 \cos ...
1
vote
1answer
25 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 ...
3
votes
1answer
24 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 ...
0
votes
1answer
43 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 ...
0
votes
1answer
31 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 ...
0
votes
1answer
31 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 ...
0
votes
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 ...
1
vote
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 ...
0
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 $A$ ...
0
votes
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?
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
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 ...
0
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 $(.)$.
0
votes
1answer
43 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 ...
1
vote
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 ...
1
vote
1answer
25 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
17 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
42 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
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 ...
0
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 ...
3
votes
1answer
45 views

An exercise of positive element in C*-algebra

Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
1
vote
1answer
40 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
votes
1answer
36 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^{**}: ...
1
vote
1answer
33 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
46 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
58 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 ...
0
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
84 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 ...
1
vote
1answer
46 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 $a \geq 0$?
1
vote
1answer
36 views

A theorem about a tracial state in von Neumann algebra

I am reading a book about C*-algebra. There is a quotation below. Let $M$ be a von Neumann algebra with a faithful normal tracial state $\tau$ and let $1_{M}\in N\subset M$ be von Neumann subalgebra. ...
2
votes
1answer
54 views

A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
0
votes
1answer
40 views

The proof of Stinespring dilation

(Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there exist a Hilbert space $H_{1}$, and a *-representation $\pi: A \rightarrow ...
0
votes
1answer
18 views

The comprehension of Stinespring dilation in C*-algebra

(Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there exist a Hilbert space $H_{1}$, and a *-representation $\pi: A \rightarrow ...
1
vote
1answer
50 views

A simple question about positive element in C*-algebra

I am reading a book about C*-algebra. There is a quotation below. An $operator~system$ $E$ is a closed self-adjoint subspace of a unital C*-algebra $A$ such that $1_{A}\in E$. The $n \times n$ ...
0
votes
1answer
34 views

A proposition in C*-algebra

Problem: Let $A$ and $B$ be C*-algebra and $\varphi:A \rightarrow B$ be a contractive completely positive map. $A_{\varphi}=\{a\in A: \varphi(a^{\ast}a)=\varphi(a)^{\ast}\varphi(a)$ and ...
1
vote
1answer
57 views

The excision theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". I can not understand the proof of the excision theorem in the fundamental facts of the book. Theorem 1.4.10(Excision) Let ...
3
votes
1answer
54 views

An exercise in C*-algebra

Let $A$ be a C*-algebra, $\phi$ be a pure state and $L=\{a\in A:\phi(a^{\ast}a)=0\}$, how to prove that $L+L^*\subseteq ker\phi$. ($L^*=\{a^{\ast}: a\in L\}$) I think it is an easy exercise, ...
2
votes
1answer
49 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
1
vote
1answer
45 views

A theorem about approximate units in C*-algebra

I am reading a book about C*-algebra. I encounter a theorem without proof. Could someone help me to complete its proof or give me some hints. Theorem 2.1. Let $I$ be an ideal of C*-algebra $A$. Then ...