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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Bott projection

This is part of an exercise problem (5.I) in Wegge-Olsen's book "K-theory and $C^*$-algebras". There he defines the Bott projection for $\mathbb{R}^2$ by $B:\mathbb{R}^2\rightarrow\mathbb{M}_2$, ...
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242 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$|\omega(A)|^2\leq\|\omega\|\cdot\omega(A^*A)$$
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84 views

The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
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Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...
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*-representations of dense subalgebras

Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a ...
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Minimal and maximal unitization of $C^{*}$ algebras

Is there a non unital $C^{*}$ algebra $A$ for which the multiplier algebra $M(A)$ is isomorphic to the minimal unitization $\tilde{A}$?
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Matrix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $$A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
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What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
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What are non-tagential limits?

I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions ...
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28 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
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32 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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31 views

The exactness of a C*-algebra

Here is a quotation: Corollary 3.7.12 If $\Gamma$ is a non-amenable residually finite group, then $C^{*}(\Gamma)$ is not exact. It follows from this corollary that $B(l^{2})$ is not exact ...
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A simple description of $ {C^{*}}(\Gamma) \otimes_{\sigma} {C^{*}}(\Gamma) $ when $ \Gamma $ is finite.

Problem. Let $ \Gamma $ be a discrete group. Denote its full group $ C^{*} $-algebra by $ {C^{*}}(\Gamma) $. If $ \Gamma $ is a finite group, then is it true that $ {C^{*}}(\Gamma) \odot ...
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A definition of discrete group

Definition: A discrete group $\Gamma$ is called residually finite if there exist subgroups $\Gamma\supset\Gamma_{1}\supset\Gamma_{2}\supset...$ such that each $\Gamma_{i}$ is a finite-index, normal ...
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56 views

The commutant of reduced C*-algebra of a discrete group

For a discrete group $\Gamma$ we let $\lambda: \Gamma \rightarrow B(l^{2}(\Gamma))$ denote the left regular representation and $\rho$ denote the right regular representation. The reduced C*-algebra of ...
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An extension of representation

Let $A,~B$ be two C*-algebras, if $A$ is an ideal in $B$, then do we have that any representation of $A$ can extend to a representation of $B$?
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32 views

The norm on tensor product

Here is a quotation of a book: Let $B, ~C$ be unital C*-algebras and $A$ be a nonunital C*-algebra, $\|\cdot\|_{\alpha}$ be a C*-norm on $B\odot C$ (the tensor product) and $\|\cdot\|_{\beta}$ be ...
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54 views

The quotient embedding of tensor product

Here is a quotation of a book: Let $A, B$ be two $C^*$-algebras and $J\subset A$ be a $C^*$-subalgebra, then there is a dense embedding $$\frac{A\odot B}{J\odot ...
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Exact sequence of tensor product

Here is a quotation of a book: Proposition 3.7.1. If $0 \rightarrow J \rightarrow A \rightarrow (A/J)\rightarrow 0$ is an exact sequence, then for every $B$, the natural sequence $$0 \rightarrow ...
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A simple question about Lance's weak expectation property.

Here is a quotation of a book: Definition 3.6.7. A C*-algebra $A\subset B(H)$ is said to have Lance's weak expectation property (WEP) if there exists a u.c.p map $\Phi: B(H)\rightarrow A^{**}$ ...
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34 views

A equivalent proposition of contractive completely positive map

Proposition 3.6.6. Let $A\subset B$ (C*-algebras) be an inclusion. Then the following are equivalent: (1). there exists a c.c.p.(contractive completely positive) map $\phi: B\rightarrow A^{**}$ such ...
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A proof of a proposition of tensor product

Proposition 3.6.5.(The Trick) Let $A\subset B$ and $C$ be C*-algebras, $||.||_{\alpha}$ be a C*-norm on $B\odot C$ and $||.||_{\beta}$ be the C*-norm on $A\odot C$ obtained by restricting ...
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Groupoid $C^*$ algebra of product groupoid

Let $G$ and $H$ be locally compact (Hausdorff, second countable) groupoids with Haar systems $\mu$ and $\nu$, respectively. Is it true then that the (full) groupoid $C^*$-algebras satisfy $$ ...
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(nx(gradxn))^2 operator question?

by $A\times B \times C = (A \cdot C)B-(A \cdot B)C$, i need to expand $n \times \bigtriangledown \times n$, where all of these are vectors. Here is what i have right now $n \times \bigtriangledown ...
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$E(a)=0\Longrightarrow E(a^{n})=0$?

Let $(M; \tau)$ be the hyperfinite $II_{1}$-Factor and consider a W${}^{\ast}$-subalgebra, $N$. Is there a (trace-preserving) conditional expectation, $E:M\to N$? Considering, now, a more general ...
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39 views

The hereditary subalgebra

If $B$ is a C*-algebra and $A\subset B$ is a hereditary subalgebra, then , taking $\{e_{n}\}$ be the approximate unit of $A$, can we verify $e_{n}be_{n} \in A$ for every $b\in B$?
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79 views

The point-ultraweak convergence of contractive completely positive map

Let $A$ and $C$ be C*-algebras. If $\phi_{n}: A \rightarrow C$ is a c.c.p (contractive completely positive) map, then the point-ultraweak cluster point of the map $\phi_{n}$ is still a c.c.p. map? ...
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Completely bounded map and minimal tensor products

Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p.(completely positive) maps. Then the algebraic tensor product map $$\phi\odot\psi: ...
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The proof of (continuity of tensor product maps) theorem

Here is a proof of (continuity of tensor product maps) theorem: Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p. maps. Then the algebraic ...
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The restriction homomorphism

Let $||.||_{\alpha}$ be a C*-norm on $A\odot B$, $A\otimes_{\alpha} B$ be the completion and $\xi$ be a state on $A\otimes_{\alpha} B$. We definte the restrictions $\xi|_{A}$ and $\xi|_{B}$ as ...
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An exercise about minimal norm

Exercise 3.4.1. Let $\pi: A \otimes B \rightarrow C$ be a $*$-homomorphism which is injective when restricted to $A\odot B$. Show that $\pi$ must be injective on all of $A\otimes B$. Is this still ...
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The unitarily equivalent between two representations

Here is a quotation of a book: Let $\phi$ and $\psi$ be the faithful states on $A$ and $B$ respectively, and let $||.||_{\alpha}$ be any C*-norm on $A\odot B$ (algebraic tensor product). As we know, ...
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The tensor product of $M_{n}(\mathbb{C})$

There is a quotation below: Let $\{e_{i,j}\}_{1\leq i, j\leq n}$ be a system of matrix units fro $M_{n}(\mathbb{C})$ and consider $$\sum\limits_{i,j=1}^{n}e_{j, i}\otimes e_{j, i}.$$ A ...
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The restriction of representation of $A\otimes_{\alpha} B$

Let $||.||_{\alpha}$ be a C*-norm on $A\odot B$, $A\otimes_{\alpha} B$ be the completion and $\xi$ be a state on $A\otimes_{\alpha} B$. Let ($\pi_{\xi}, H_{\xi}, v_{\xi}$) be the GNS triplet and ...
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59 views

Several questions about state space

Here are several questions about the state space of a C*-algebra $A$: Let $A$ be a unital and separable C*-algebra, can we find a faithful state $\phi \in S(A)$. ( The $S(A)$ denotes the state space ...
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Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
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Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
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The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing ...
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70 views

A lemma about the pure states

There is a quotation of a book: Lemma 3.4.5. Assume that both $A$ and $B$ are unital and abelian C*-algebras. Then for every C*-norm $\|\cdot\|_{\alpha}$ on $A\odot B$ and pair of pure states ...
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37 views

The GNS of a pure state on $A\otimes_\alpha B$

Let $A, B$ be the C*-algebras and $\|\cdot\|_\alpha$ be a C*-norm on $A\odot B$, $\xi$ be a state on $A\otimes_\alpha B$. (Here, $\odot$ denotes the algebraic tensor product and $A\otimes_\alpha B$ ...
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58 views

How to verify the isomorphism between two C*-algebra

Let $B$ be a C*-subalgebra of a unital C*-algebra $A$, how to verify $C^{*}(B, 1_{A})\cong \tilde{B}$? Here, $C^{*}(B, 1_{A})$ denotes the C*-algebra generated by $B$ and $1_{A}$, meanwhile the ...
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An interesting phenomenon of $C^*$-tensor product

On the algebraic tensor product space of $C^*$-algebra, I try to find an example whose maximal $C^*$-norm is not the minimal $C^*$-norm, but it seems as it is impossible to do this because the finite ...
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Questions about multiplier algebra and corona algebra

When I read N.E. Wegge-Olsen's book K-theory and C-star-algebras_ A friendly approach I meet the following two problems about standard isomophisms: For any $C^\ast$-algebra $\mathcal{A}$, is ...
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application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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Polynomial Ring: Root vs. Remainder

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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194 views

Question about finite rank operators

Let $X$ be a normed space, $\mathcal{F}(X)$ the algebra of all operators on $X$ with finite fank, then $\mathcal{F}(X)$ is the unique minimal ideal of $\mathcal{K}(X)$ the algebra of all compact ...
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an invariant of $C^{*}$ algebras

consider the following property (invariant) for complex $C^{*}$ algebras: "$T(x)=x^{*}$ is the only non zero $\mathbb{R}$-linear map on $A$ which satisfies $T(x)T(y)=T(yx)$." Questions: 1)Some ...
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The commutative tensor product norm

Definition 3.3.3 (Maximal norm) Given $A$ and $B$, we define the maximal C*-norm on $A\odot B$ to be $$||x||_{max}=sup\{||\pi(x)||:\pi:A\odot B\rightarrow B(H) a *-homomorphism\}.$$ for $x\in A\odot ...
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The state on C*-algebra

Let $C$ be a C*-algebra, $A\subset C$ be a C*-subalgebra of $C$ and $B=A'\cap C$ (here, $A'$ denotes the commutant of $A$). If $\xi$ is a state on $C$ and we take an positive element $b\in B$, then ...
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100 views

An application of Hahn-Banach (separation) theorem

Here is a quotation of a book: Let $S(A)$ denote the state space of a C*-algebra $A$ and $M\subset S(A)$ denote a weak-$*$ closed convex set. Assume there is a state $\psi$ which does not belong ...