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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Comparison of positive elements and Hilbert C*-modules

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras. Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
2
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1answer
116 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
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1answer
87 views

characters of a $C^*$-algebra

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
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1answer
67 views

When is the image of a GNS representation WOT-dense?

Given a $C^*$-algebra $A$ and a state $\rho$ on $A$, let $\pi_\rho$ be the corresponding GNS representation on the Hilbert space $H_\rho$. I would like know when the image of $\pi_\rho$ is WOT-dense ...
4
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1answer
183 views

Quotients of C*-algebras

It is known that every unital separable C*-algebra is a quotient of the full group C*-algebra $C^*(F_I)$, where $F_I$ is the free group generated by some index set $I$. Can we drop the ...
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210 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
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1answer
115 views

Quotients of the maximal tensor product

Let $A$ and $B$ be C*-algebras and let $\gamma$ be any C*-norm on the algebraic tensor product $A\odot B$. Why is $A\otimes_\gamma B$ a quotient of $A\otimes_{{\rm max}}B$, where $\otimes_{{\rm max}}$ ...
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153 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
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123 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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1answer
63 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
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1answer
94 views

Definite states on C*-algebras

A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$. I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
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98 views

On the Spectral Theorem

Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure. a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
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186 views

States and positive elements in $C^*$-algebras

Let $A$ be a unital $C^*$-algebra and $w$ be a state (i.e a positive linear functional such that $\|w\|=w(1_A)=1$. I'm trying to prove the following:a) if $a$ is selfadjoint and $w(a^2)=w(a)^2$ then ...
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1answer
78 views

A question concering nuclearity

B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product ...
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87 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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1answer
218 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
2
votes
2answers
213 views

Unitary operator in von Neumann algebra

Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
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1answer
67 views

What is $\sigma_{\mathfrak{M}_{p}}(p\cdot a\cdot p)$ when $p=\chi_{S}(a)$ for closed $S\subseteq \sigma(a)$?

Let $\mathfrak{M}$ be a von-Neumann algebra and $a\in\mathfrak{M}$ a self-adjoint element. Let $S\subseteq\sigma(a)$ be a non-empty, closed subset. Let $p=\chi_{S}(a)$, which is a non-zero, ...
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1answer
67 views

differential equation of the square root of a matrix

If the differential equation governing the time dependent matrix $M(t)$ is $\frac{dM(t)}{dt}=A.M(t).B$ or $\frac{dM(t)}{dt}=A.M(t)+M(t).A$ where $A$ and $B$ are constant matrices, what is the ...
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227 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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47 views

Are decomposable maps completely bounded?

By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose ...
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36 views

A map that is $(n-1)$-positive but not $n$-positive

Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
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141 views

How to prove this element is strictly positive?

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
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1answer
104 views

Is there any operator which its spectrum corresponding to a compact set?

we know that for each operator $T$ the spectrum $\sigma(T)$ is compact. Is the converse true I mean if we have a compact set $K\neq\emptyset$, is there any operator $T$ such that $\sigma(T)=K$? I am ...
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1answer
47 views

Are self-adjoints elements norming?

Let $\mathsf A$ be a C*-algebra and let $\mathsf{A}^*$ be its dual space. Is it true that for $f\in \mathsf A^*$ we have $$\|f\|=\sup\{|f(x)|\colon\; \|x\|=1\mbox{ and }x^* = x\}?$$
3
votes
2answers
83 views

ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
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1answer
96 views

strictly positive in unital $C*$ algebra

Let $A$ be a unital $C*$ algebra. $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertable. I proved one direction : if $a$ is ...
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38 views

Finding Strictly Positive Elements [duplicate]

I need to find the set of strictly positive elements in the $C^*$-algebra $C_0(\Omega)$ where $\Omega$ is a locally compact Hausdorff space. Clearly, the set will be contained in $ \{ f \in ...
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2answers
161 views

Positive elements in $C^*$-algebras

I'm trying to prove the following, and I'm not sure if the proof is correct? If $A,B$ are $C^*$-algerbas, and $f$ is a $*$-homomorphism from $A$ onto $B$ then $f(A_+)=B_+$.Proof: let $a\in A_+$ then ...
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votes
2answers
128 views

example of positive but not completely positive operator

I was looking for some example of a positive operator which is not completely positive on a banach algebra. if I consider my banach algebra to be $\text{M}_n(\mathbb{C})$ of matrices over complex ...
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2answers
49 views

Structure of $L_1(G)$

I came across this while going through some basic examples of $C^*$ algebras. If I consider $G$ as the set of cube roots of unity, what will be $L_1(G)$? I mean what will be the structure of elements ...
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1answer
304 views

Cube root in $ C^{*}$-algebra.

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
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0answers
68 views

2 positive decomposable maps

A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
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votes
1answer
174 views

Determining whether an operator is Hermitian

The operator $F$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a) $, where $a$ is a nonzero constant. Determine whether or not $F$ is a Hermitian operator. If the condition for $F$ to be Hermitian is ...
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2answers
121 views

Recovering a group from its C*-algebras and group algebra

Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions? Is it true that: if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$? if $C_r^*(G)$ and ...
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votes
1answer
47 views

Help proving operator inequality

Given $P \geq 0$, I need to show that $2Tr(P^{5/2}) \leq Tr(P^3) + Tr(P^2)$. It's trivial to show that the RHS is the trace of a positive operator, but I'm at a loss on how to actually prove this ...
2
votes
2answers
93 views

Multiplicative linear functional on algebra of limit of polynomials

Let $A$ be the space of all functions which are limit of polynomials over the unit ball $D$. Then $A$ is a commutative Banach algebra. Then how do I show that $A$ has no non zero multiplicative linear ...
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1answer
92 views

Linear functional on Banach algebra

Let $A$ be the space of all matrices of the form $\begin{pmatrix} a & b \\0 & a\end{pmatrix}$, $2\times2$ over complex field. Then the spectrum of any element of $A$ comes out to be $\{a\}$. I ...
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139 views

Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
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votes
0answers
109 views

Operator monotone functions

By definition, I know that a function $f$ is operator monotone if $A - B \geq 0 \Rightarrow f(A) - f(B) \geq 0$. For instance, we have $A^2 \leq B^2 \Rightarrow A \leq B$ because the root function is ...
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1answer
76 views

Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?

It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
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61 views

problem related to tensor product on Hilbert spaces

Let $K$ and $H$ be Hilbert spaces. Let $\{e_i:i\in I\}$ be an orthogonal basis of $H$. Define $$ U_i:K\to K\overset{.}{\otimes} H: x\mapsto x\overset{.}{\otimes} e_i $$ Assume ...
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1answer
173 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
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votes
1answer
283 views

Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
3
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1answer
105 views

Counterexample for a polar decomposition in von Neumann and $C^\ast$ algebras

For a von Neumann algebra, we have that partial isometry and positive operator of an operator in its polar decomposition belongs to the algebra, but in a $C^\ast$ algebra this may not be true. Can ...
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1answer
155 views

State space is weak* compact

I'm trying to convince myself that the state space $S(A)$ of a unital $C^*$-algebra is weak* compact. I've proven that $S(A)$ is convex, and I feel that this should allow me to conclude weak* ...
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1answer
59 views

Set of states not compact

I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course). I think $K(H)$, the compact operators over a Hilbert space ...
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1answer
150 views

Multiplication not continuous in $B(H)$ in the strong operator topology

I'm reading this answer by t.b., and I'm only interested in the case when $X$ is an infinite-dimensional Hilbert space. Regarding Question 2, in the first bullet point he claims the following: "Given ...
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122 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
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1answer
59 views

Why should we use inverse homemorphism here?

I am a brand-new comer in dynamical system. I find it interesting that when defining ergodicity of classical dynamical system $(X,\sigma)$, they use $\mu(\sigma^{-1}(E))$ there. Since $\sigma$ is a ...