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)

3
votes
1answer
53 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
25 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
65 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
51 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
67 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
26 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
48 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
31 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
57 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
51 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
35 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
36 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
62 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
48 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
36 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
44 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 ...
1
vote
1answer
31 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
90 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
48 views

The matrix in C*-algebra

Let $A$ be a C*-algebra and $A^{**}$ be the double adjoint of $A$. Can we conclude $M_{n}(A^{**})\cong (M_{n}(A))^{**}$?
3
votes
1answer
72 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
2
votes
1answer
75 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
55 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
58 views

The spectrum of the operators

Let $X, Y$ be the Banach space, and $T_{1}: X\rightarrow X$ and $T_{2}: Y\rightarrow Y$ be the bounded linear operators. Then what is the relationship between $\sigma(T_{1})$, $\sigma(T_{2})$ and ...
2
votes
1answer
56 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
48 views

The norm of operator matrix

Let $H$ be a Hilbert space and $B(H)$ be the bounded linear operator on $H$, for $T\in B(H)$, if $T=\left(\begin{array}{ccc} 0 & B \\ A & 0 \\ \end{array}\right)$ on $H=M\oplus ...
0
votes
1answer
49 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 ...
1
vote
1answer
69 views

The operator matrix on Hilbert space

Let $H$ be a Hilbert space and $P$ be the projection operator, then $H= P(H)\oplus (1-P)(H)$. Hence, for each $T\in B(H)$, we have $$T=\left(\begin{array}{ccc} PTP & PT(1-P) \\ (1-P)TP ...
3
votes
1answer
74 views

A easy question on projection operator

Let $H$ be a Hilbert space and $B(H)$ be all the bounded linear operators on $H$, for arbitrary $T\in B(H)$, if $\{P_{i}\}$ is an increasing net of finite-rank projection, can we conclude $P_{i}TP_{i} ...
3
votes
1answer
115 views

Projection operator in Hilbert space

Let $H$ be a Hilbert space, can we find an increasing net of finite rank projections which converge to the identity in the strong operator topology? And I think if $H$ is separable, we can find an ...
0
votes
1answer
75 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 ...
1
vote
1answer
51 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
86 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
102 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 ...
4
votes
0answers
87 views

Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
1
vote
1answer
97 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
168 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 ...
4
votes
1answer
52 views

continuous depence of the spectrum on elements

Suppose $a_n \to a$ in a unital $C^*$-algebra $A$. If $\lambda_n \in \sigma(a_n)$ converges to $\lambda \in \mathbb{C}$, then $\lambda \in \sigma(a)$. Does the converse hold? So if $\lambda \in ...
0
votes
0answers
82 views

Abelian group C*-algebras

Let G is a locally compact Abelian group $C^*$-algebra, then $C^*(G)$ is an Abelian $C^*$-algebra, so C*(G) is isomorpohism with the C$_0$(X) for some locally compact Hausdorff space X, here X is the ...
1
vote
1answer
77 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
39 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
26 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
52 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 $a \geq 0$?
0
votes
1answer
26 views

Has a *subalgebra of $L(H)$ with $1$ a trivial null space?

Like the title: is it true that a self-adjoint unital subalgebra of $L(H)$ closed in the weak operator topology (a Von Neumann algebra) has a trivial null space? Why? Thank you.
1
vote
1answer
141 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
204 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 ...
2
votes
1answer
36 views

Does it hold that $\{a\}'' = \{f(a) : f$ bounded, Borel measurable$\}$?

Let $H$ be an Hilbertspace. Let $a\in\mathcal{B}(H)$ be a self-adjoint operator. Does it hold that $\{a\}^{\prime\prime}=\{f(a):f\in L^{\infty}(\sigma(a))~\text{Borel measureable}\}$?
5
votes
0answers
34 views

freely generating elements in an algebra

Let $(\mathfrak{M}, \tau)$ be a W${}^{\ast}$-Algebra with (finite, normal, etc., whichever nice conditions one may find need for) tracial state. Elements $(a_{i})_{i\in I}\subset\mathfrak{M}$ shall be ...
0
votes
1answer
61 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
50 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
106 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 $n \times n$ ...