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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Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
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1answer
40 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
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36 views

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$.

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$. How to construct this example? I have obtained that of ...
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69 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
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67 views

Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible

Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$. How can one show that if the commutant has ...
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72 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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132 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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96 views

Spectrum in Banach Algebra

Let $A$ be a unital Banach algebra and $a\in A$. Let $U$ be an open subset of $\mathbb C$ containing $\sigma (a)$. Prove that there is $\delta>0$ such that for every $b\in A$, if ...
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179 views

Extension of Choi's theorem on extreme completely positive maps

In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below. Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is ...
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97 views

Decomposition of representations

Let $A$ be a (possibly nonunital) Banach *-algebra, and $H$ be a Hilbert space. If $\pi: A \to B(H)$ is a *-homomorphism, i.e. a representation, then why must $\pi$ be equivalent to a direct sum of ...
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99 views

A question about ideals in operator algebras

Is it true that in a unital C*-algebra $A$ every closed left ideal $L\subset A$ is an intersection of all maximal left ideals which contain $L$? I know that $L$ is the left-kernel of some state but I ...
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1answer
247 views

Ultraweak topology

In Stratila and Zsido, as well as some other sources, the ultraweak topology on $B(H)$ is taken to be the smallest topology for which every element in the closure of the span in $B(H)$ of the elements ...
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1answer
118 views

Subprojection of a finite projection

The question is entirely explained here, in that I wonder why every source seems to regard as obvious the claim that subprojections of finite projections are finite. Here is the link. To me, playing ...
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1answer
138 views

Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
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1answer
70 views

Classification of Type 1 factors

In the proof of this theorem, which says all of the type 1 factors (factors with minimal projections) are isomorphic to $B(\ell^2(I))$ for some $I$, I want to know a few things: The supposed ...
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134 views

Generation of Von Neumann Algebras

Suppose $M$ is a Von Neumann Algebra. (VNA) For me, these are subsets of some $B(H)$ that are $*$-algebras, containing the $1$ of $B(H)$, that are Weak Operator (WO) closed, or equivalently Strong ...
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105 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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Space of operators on function

Consider the following space of operators on function of $n$-variables $A= Span \{x_ix_j\ , x_i \frac{\partial}{\partial x_j} , \frac{\partial^2}{\partial x_i \partial x_j} , i,j=1,2,\cdots,n\}$. ...
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58 views

A question about positive normal linear functionals in C*-algebra

There is a question below: Let $\phi :B(H) \rightarrow M_{n}(C)$ be a contractive completely positive map. Since $\phi$ is a contractive completely positive map, the corresponding functional ...
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22 views

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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Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
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30 views

What is Kadison's process about cocycles?

My teacher told me the Kadison's process(may be not this ward, it is just my translation ) can make a 2-cocycle turn to be a cocycle(i.e.,derivation). But I can not find it in the internet. Thanks a ...
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Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
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Does an irreducible operator generate a nuclear $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible (Halmos 1968) if its commutant $\{ T\}'$ ...
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57 views

Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
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1answer
76 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
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24 views

Factorizing a saddle point operator

I have a coupled pde which, after semi-discretizing in time, results in the solution of a sequence of continuous saddle point problems $Lu^t=b$. Written explicitly, the problem looks like this: ...
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1answer
51 views

Stable group algebras

Let $G$ be a discrete group and let $C^*_r(G)$ be its reduced group C*-algebra. Is there any group $G$ for which we have $C^*_r(G)\cong M_2(C^*_r(G))$? Or more generally, $C^*_r(G) \cong ...
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Relationship of two generalizations of the real/complex calculus

On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ...
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Cap product between K-Theory and K-Homology

In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative. Is the commutativity ...
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1answer
105 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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41 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 ...
2
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1answer
71 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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150 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
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93 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
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45 views

Quotients of the CAR algebra

Recently, I heard about the following theorem: each nuclear separable operator space is a completely bounded quotient of the CAR algebra. Yet, I have no idea who and where proved this theorem ...
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80 views

Sign problem with Poisson brackets

I am wondering if anyone could explain to me either why my method is not valid or point out where I have made an algebraic slip. I have been looking at this for a long time, to no avail. Let $\{\cdot ...
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115 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
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74 views

Induced representations of topological groups

Sorry if this is a naive question-- I'm trying to learn this stuff. If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to ...
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51 views

Question considering masas

Suppose $M_1, M_2$ are type II_1 factors, A is a masa in $M_1\otimes M_2$, can we find a *-automorphism $\phi$ on $M_1\otimes M_2$ such that there are two masas $A_1\subset M_1,A_2\subset M_2$ and ...
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2answers
53 views

A question about a linear bounded operator (in hilbert space)

I am reading a book about C*-algebra. When i study von Neumann algebras in this book, i meet with a problem. In the book, If $H$ is a Hilbert space, we write $H^{(n)}$ for the orthogonal sum of n ...
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1answer
47 views

Question about operators on Hilbert space

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for ...
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2answers
39 views

A question about discrete group

There is a quotation below: For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, ...
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1answer
31 views

Commutator proof

I have a proof in my book I dont fully understand. The author is proving that if $[A,B]=1$ then $[A,B^n]=nB^{n-1}$. The proof is really short, it is only one line of equations, but I dont understand ...
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2answers
59 views

Example of a net in $\mathcal{B}(\ell_2)$ that converges in the weak operator topology but not in the strong operator topology?

I need to show that the Strong Operator topology is strictly stronger in $\ell_2$ (space of complex sequences that are square summable). I can show that convergence in the strong operator topology ...
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2answers
35 views

Is each element of factor(von Neumann algebra) a linear combination of projection?

Let $A$ be a factor von Neumann algebra. Then every element of $A$ can be writeen as a finite linear combination of projections in $A$. Is it right? I know a little about von Neumann algebra, thanks a ...
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91 views

The free group $II_{\infty}$ factor isomorphism problem

Let $\Gamma$ be an infinite discrete group, and $H = l^{2}(\Gamma)$ the separable infinite dimensional Hilbert space. Let $\rho$ be the left regular representation of $\Gamma$ on $H$. Definition : ...
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2answers
118 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
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1answer
106 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* ...