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 ...

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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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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 ...
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40 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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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^*\| = ...
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54 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 ...
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46 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 ...
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48 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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33 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 ...
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92 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 $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
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49 views

The matrix in C*-algebra

Let $A$ be a C*-algebra and $A^{**}$ be the double adjoint of $A$. Can we conclude $M_{n}(A^{**})\cong (M_{n}(A))^{**}$?
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76 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 ...
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81 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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57 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 ...
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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 ...
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57 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 ...
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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 ...
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52 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 ...
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71 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 ...
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75 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} ...
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127 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 ...
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83 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 ...
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59 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 ...
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92 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 ...
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108 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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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 ...
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107 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. ...
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197 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 ...
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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 ...
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97 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 ...
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82 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 ...
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42 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 ...
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30 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: ...
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58 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 $a \geq 0$?
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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.
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169 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. ...
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248 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 ...
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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 $\{a\}^{\prime\prime}=\{f(a):f\in L^{\infty}(\sigma(a))~\text{Borel measureable}\}$?
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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 ...
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69 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 ...
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55 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 ...
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112 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 $n \times n$ ...
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43 views

An exercise in operator theory

Let $H$ be a Hilbert space and $P$ be a projection to a finite dimensional subspace $K$ of $H$, for a $T\in B(H)$, if $||PTP||=1$, then, for arbitrary $\epsilon>0$, there exists a vector $\alpha$ ...
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Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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Concrete representation of the annihilating algebra

Suppose $\mathfrak{M} = A^{\prime\prime}$, where $A$ is a concretely described subalgebra of $\mathcal{B}(\ell^{2}(\mathbb{N}))$. In some instances, it is possible to provide a concrete description of ...
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40 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
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43 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 ...
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83 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 ...
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77 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, ...
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42 views

A question about $C^\ast$-algebra

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a ...
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100 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...