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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Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
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209 views

Extreme points in the set of positive linear functionals of norm $\leq 1$

Let $A$ be a C* algebra, and $S$ the set of positive linear functionals on $A$ in the unit ball of $A^*$ (Which has the weak-* topology.) I am having difficulty seeing that all nonzero extreme points ...
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222 views

One particular application of the Cauchy Schwarz Inequality

A document I am reading on Von-Neumann algebras (VNA) asserts that it follows from Cauchy-Schwarz that if $M$ is a VNA, and $w$ is a positive linear functional on M that is merely norm continuous, ...
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120 views

Complemented ideals in von Neumann algebras

Let $I$ be an ultraweakly closed ideal in a von Neumann algebra $M$. For example, this can be the kernel of an ultraweakly continuous homomorphism. Is it true that there is another ideal $J\subset M$ ...
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35 views

A detail in Rădulescu's Theorem proof

I've been following one Rădulescu's Theorem proof ($G$ is hyperlinear if and only if $\mathcal{L}_{G}$ can be embedded in a ultrapower of the hyperfinite type-$II_{1}$ factor $\mathcal{R}$, where ...
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150 views

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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51 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^*\| = ...
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35 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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87 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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64 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
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164 views

GNS-triplets for states on the matrix space: generalization to the infinite-dimensional setting

In a previous exercise, I have proven that states $\omega$ on the $C^*$-algebra $M_n(\mathbb C)$ correspond to a unique density matrix $\rho$ by the relation $\omega(A) = \mathrm{Tr}(\rho A)$. I was ...
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62 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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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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58 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 ...
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184 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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137 views

Entirely “Bare-hands” proof that completely additive states are ultraweakly continuous

Originally I had asked this question: Positive Linear Functionals on Von Neumann Algebras I got some responses that directed me to a variety of resources, some of which I could not understand because ...
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361 views

Spatial tensor product of algebras and normed spaces

Let $A_1$ and $A_2$ be two $C^*$-algebras considered as closed $*$-subspaces of some $\mathcal{B}(H_1)$ and $\mathcal{B}(H_2)$. More preciesly there exist faithfull $*$-representations ...
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40 views

Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here ...
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33 views

Understanding minimal projections

This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition. Let $A$ be a C*-algebra and let ...
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45 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
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Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
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144 views

application of c*algebras

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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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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145 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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133 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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72 views

maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
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Given $\theta(p)\neq p$ does there exist $q\leq p$, so that $\theta(q)q=0$?

Let $\mathfrak{M}$ be a vN-Alg. Let $\theta\in \text{Aut}(\mathfrak{M})$. Let $p\in\mathfrak{M}$ be a projection, so that $\theta(p)\neq p$. Is there a projection $q\in\mathfrak{M}$ with $0\neq q\leq ...
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103 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
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138 views

Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
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Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
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144 views

Basis for completely bounded maps.

The set of completely bounded (CB) maps forms can be considered as a complex span of the set of completely positive (CP) maps. Can we find a basis for this complex linear space of CB maps such that ...
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1answer
180 views

reduced crossed products

Given a discrete group $G$ and a $G$-$C^*$-algebra $A$ we can form the reduced crossed product $A\rtimes_r G$. I want to define it by the closure of $C_c(G,A)$ in $\mathcal{B}(\ell^2(G,A)$ where this ...
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Dropping homomorphisms to quotients of C$^*$-algebras

Let $A$ be a C$^*$-algebra, let $\Delta:A\rightarrow A \otimes_{\min} A$ be a $*$-homomorphism, and let $\phi$ be a state on $A$. Let $(H,\pi,\xi_0)$ be the GNS construction for $\phi$; let $B=\pi(A) ...
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263 views

When does Stinespring dilation yield a faithful representation?

Let $A$ be a $C^*$-algebra, $H$ a Hilbert space, $\phi: A \to B(H)$ a completely positive map. The Stinespring construction yields a triple $(K, V, \pi)$ where $K$ is a Hilbert space, $V: H \to K$ a ...
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424 views

Eigenvalues of operators

I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way: $T(A)=0.5(A-A^\mathrm{t})$ ($A^\mathrm{t}$: the transpose of a matrix $A$). I need to prove ...
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109 views

Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
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3answers
56 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
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2answers
81 views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
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170 views

universal C* algebras

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would ...
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2answers
167 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
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2answers
69 views

Positive elements of a $C^*$-algbera form a poset

My knowledge of $C^*$-algebras is very little. We call an element positive if $a=b^*b$ for some $b$ and make a relation on all positive elements by saying $$ b \geqslant a \iff b-a \text{ is ...
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An exercise (about positive elements) in C*-algebra

Let $A$ be a C*-algebra, $a\in A$ be a positive element and $b\in A$ be an arbitary element in $A$. Can we verify that $$b^{*}ab\leq \|b\|^{2}a~~?$$
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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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Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
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1answer
63 views

Irreducible representations and commutative C*Algebras.

If $A$ is a commutative C*-Algebra then also its representation $\pi(A)$ is commutative, and it's an operator C*-algebra. A representation is said to be irreducible if $\pi(A)$ does not commute with ...
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A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
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1answer
62 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
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58 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
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208 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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52 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...