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 routine exercise about matrix norm

If $T_{n}\in M_{k(n)}(\mathbb{C})$ and $||T_{n}^{*}T_{n}-1_{k(n)}||\rightarrow0$, then $||T_{n}T_{n}^{*}-1_{k(n)}||\rightarrow 0$ too? (Here, $M_{k(n)}(\mathbb{C})$ denotes the $k(n) \times k(n)$ ...
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52 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
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23 views

A question about Invariant subspaces of an algebra.

I feel that this is a very simple problem, but somehow I don't see the solution. I want to show that if $A$ is a subalgebra of $B(H)$ containing $1$ then if $B\in SOTcl(A)$, for every n, ...
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40 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
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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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98 views

Properties of a $B^\ast$-algebra

Defining a complex Banach algebra as a $B^\ast$-algebra when it is equipped with an application $\ast:B\to B$ such that for any $x,y\in B$ $$(x+y)^\ast=x^\ast+y^\ast,\quad(xy)^\ast=x^\ast ...
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164 views

Finding the norm of this upper triangular matrix

I have a matrix $A=\begin{pmatrix} a & b\\ 0 & a\end{pmatrix}\in M_2(\mathbb{C})$. Given that $|a|<1$ and $|b|\leq 1-|a|^2$, I am supposed to show that $\|A\|\leq 1$ (operator norm). I ...
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63 views

A proposition about residually finite dimensional C*-algebra

Here is a proposition in a book "C*-algebra and Finite-Dimensional Approximations" P239 Proposition 7.1.8 Every type I C*-algebra with a faithful tracial state is RFD. Proof Let $\tau$ be a ...
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27 views

A question about *-homomorphism

Let $A$ be a C*-algebra and $\phi_{i}: A\rightarrow M_{k(i)}(\mathbb{C})$ (the $M_{k(i)}(\mathbb{C})$ denotes the $k(i)\times k(i)$ complex matrices) be c.c.p maps which are asymptotically ...
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38 views

A question about functional calculus

Here is a Lemma in a book "C*-algebras and Finite-Dimensional Approximations" P238. Definition 7.1.1 A C*-algebra $A$ is called quasidiagonal (QD) if there exists a net of c.c.p. maps $\phi_{n}: ...
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Diagonalization of total angular momentum over creation operators for an isotropic harmonic oscillator?

You have an isotropic three dimensional quantum harmonic oscillator so the Hamiltonian is $$ H=\frac{p^2}{2}+\frac{r^2}2 $$ If you do the creation-annihilation operator-algebra trick and define ...
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91 views

Bounded operators with infinite matrix representations

Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and that $I$ is a non-empty set. If $A\subseteq B(K)$ for some Hilbert space $K$, we can ...
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34 views

Polar decomposition in a finite von Neumann algebra

Suppose $\mathcal{M}$ is a finite von Neumann Algebra and $a \in \mathcal{M}$. Does it follow that we can write $a = |a|u$ where $u$ is a unitary operator. I know we can write $a = |a|u$ where $u$ ...
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21 views

A question about positive forms on involutive algebras.

A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$. To be useful, this definition requires that is not always possible to write ...
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46 views

An exercise about the positive operator

Here is an exercise in functional analysis: An operator $T$ on Hilbert space is positive is positive if and only if all compressions by finite-rank projections ($P_{n}TP_{n}$ for any $n$) are ...
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When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
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62 views

Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
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33 views

The uniform homeomorphism between $\mathrm{Prob}(\Gamma)$ and $l^{2}(\Gamma)$

Here is a quotation of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. In the proof of Proposition 4.4.5. (In P132), the author says: The assertion $(1)\Longleftrightarrow ...
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1answer
65 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 ...
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87 views

sums of projections in C*-algebras

Let $A$ be a $C$*-algebra with unit $e$. If $p$ and $q$ are projections such that $p+q+\lambda e$ is a projection for some $\lambda\in\mathbb{R}$, is it true that the only possible values for ...
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120 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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Strong closure of a C*-algebra of operators.

In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator ...
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A question about the definition of $(X\rtimes\Gamma)$-C*-algebra

Here is a quotation in the book "C*-algebras and Finite-Dimensional Approximations": Instead of considering the *-algebra of finitely supported functions from $\Gamma$ to $C(X)$ (C(X) denotes all ...
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108 views

Construction of noncommutative torus

In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, ...
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30 views

How to find a sequence in discrete group

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?
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87 views

What must I do with this Linear operator?

Can anyone help help me with this problem I want to show that the operator $\mathscr{L}=u \partial/\partial y$, is not an linear operator. This is what i got $\mathscr{L}(au+bv)=u \partial/\partial ...
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96 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
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64 views

Dual subfactor and commutant

Let $(N \subset M)$ be a subfactor and $N \subset M \subset M_1$ the basic construction. Question: Is $(M \subset M_1) \simeq (M' \subset N')$? Else in which generic case it's true? What's the ...
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220 views

Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
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Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
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38 views

How to prove the following map is a c.c.p map

Here is a quotation in a book "C*-algebrass and Finite-Dimensional Approximations" by Nate and Taka (P122). Let $A$ be a C*-algebra, $\Gamma$ be a discrete group and the $\alpha$ is an action of ...
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Is $vN(M_1,M_2) \cap M_3= vN(M_1,M3) \cap vN(M_2,M3)$?

Let $M_1,M_2,M_3$ be von Neumann algebras (i.e. weakly closed subalgebras of $B(H)$ where $H$ is a Hilbert space). Let $vN(M_1,M_2)$ denote the von Neumann algebra generated by $M_1$ and $M_2$ inside ...
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55 views

An exercise about reduced crossed product of a C*-dynamical system

Here is an Exercise in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If ...
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A question about cross product of C*-dynamical system

Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. ...
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57 views

Proof of operator norm with the equality of involution

Given the equality $$\|A^*A\| = \sup_{\| x \| = \| y \| = 1} | (Ax, (A^*)^*y) | $$ How do show that it is equal to $\|A\|^2$ Is it by using Cauchy-Schwarz inequality such that $(x, y) \leq ...
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62 views

Why is the Gelfand transform injective?

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
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110 views

Abelian projection in the von-Neumann algebras

As we know, a minimal projection must be Abelian, An Abelian projection must be finite. A minimal projection correspond to a rank one operator, a finite projection correspond to a finite rank ...
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39 views

Show the algebra of all bouned operators is a $W^*$-algebra

Given that the algebra of all bounded operators $B(H)$ on Hilbert space is a $C^*$-algebra such that $\| A^*A \| = \|A\|^2$. Then how can we show that it is also a $W^*$-algebra, where a $W^*$-algebra ...
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38 views

$C^{*}$ algebras positively dominated by finite dimensional algebras

Assume that $A$ is a $C^{*}$ algebra and $B$ and $C$ are two sub $C^{*}$ algebras of $A$ such that: $B$ is finite dimensional algebra. For all positive $c\in C$, there exist a positive $b\in ...
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$R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$ \exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad $$ implies $A$ is unital. I know this is true if A is a ...
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Spectrum of projection

I have a question concerning the spectrum of a projection in an abstract $C^*-$algebra. Let $A$ be $C^*-$algebra, $p\in A$ a projection (i.e. a selfadjoint idempotent). Can we say what is the spectrum ...
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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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Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
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1answer
42 views

Do two II$_1$-factors with trivial intersection generate $B(H)$?

Let $H$ be an infinite dim. separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $A$, $B \subset B(H)$ be II$_1$-factors such that $A \cap B = \mathbb{C}I$. Examples: (1) Take ...
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Where is the most clear and concise exposition of the spectral theorem for self-adjoint operators on Hilbert space?

This question is certainly subjective, which may warrant votes to close. I'm simply looking to find the "best" written exposition of the spectral theorem for possibly unbounded self-adjoint operators ...
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“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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45 views

find element in relative commutant of a matrix subalgebra

Let $M=A*P$ to be a free product von Neumann algebra, and $A$ is a finite dimensional subalgebra, for simplicity, we may assume $A=M_2(\mathbb{C})$, and $P\neq \mathbb{C}$. A standard fact is that ...
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85 views

Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
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50 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 ...