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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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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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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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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26 views

Normal states are dense in $B(H)$

Can someone explain/sketch the proof of the fact stated in the title : The set of normal states (in some $B(H)$) is weakly dense in $B(H)$ ? Or, if possible, some reference. Thank you very much.
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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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49 views

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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63 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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61 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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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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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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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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If $A$ is a primitive / irreducible C*-algebra, then $M(A)$ has trivial center.

Recall some definitions: a sub-C*-algebra $A$ of $B(H)$, the algebra of bounded operators on a Hilbert space $H$, is called irreducible if the only closed $A$-invariant subspaces of $H$ are $0$ and ...
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A question on nuclearity

Definition 2.1.1. If $A$, $B$ are C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
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20 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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48 views

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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Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
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146 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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A question on simple and unital $C^\star$-algebra

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations" Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
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139 views

A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
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61 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
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Exponential map in $C^{*}$-algebra and unitary invariance

Let $A$ be a unital $C^{*}$-algebra. Let $X$ be a closed vector subspace of $A$ which is unitarily invariant in the sense that $uXu^{*}\subseteq X$ for all unitaries $u$ of $A$. I want to show that ...
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When an invertible element in a $C^{*}$-algebra is unitary

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using ...
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99 views

An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
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181 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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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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113 views

Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want ...
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96 views

Dense subalgebras of von Neumann algebras and increasing nets

Let $N$ be a von Neumann algebra, and $A$ be a dense $*$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that: For any $x\in N^+$, there exists a increasing net $(x_j)$ in ...
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Taylor expansion for matrices

Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
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Conjugates in integrals

Is it always true that $\int f(x)^*M^\dagger Mf(x) dx=\int (Mf(x))^*(Mf(x)) dx$? where $M$ is an operator. If so, is there a simple proof that this is so? THanks.
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An elementary perturbation result in C*-algebra

The following question was raised when I read a papaer "MF actions and K-theoretic dynamics". In one of the proofs in that paper, the author utilized a so called "elementary perturbation result in ...
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QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital ...
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37 views

Infinite projections in the Cuntz algebra

I am studying the Cuntz algebra $\mathcal{O}_n$, $(n \ge 2)$ with generators $S_1, S_2, \ldots, S_n$ and in my class notes there is a statement about the projections $S_1S_1^*, S_2S_2^*, \ldots, ...
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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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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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40 views

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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39 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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Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
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1answer
27 views

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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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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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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Zhou operator theory book, Kaplanskys formula

In Zhou's operator theory book, Kaplanskys formula has stated that if $P$ and $Q$ are projection in a von neumann algebra $A$ acting on $H$, then $P\vee Q-Q\sim P-P\wedge Q$. In the proof, it says ...
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39 views

Are contractive completely positive maps trace decreasing?

Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is ...
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33 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
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35 views

The comprehension of a paragraph about point-ultraweak convergence

There is a quotation below (in the book "C*-algebras and Finite-Dimensional Approximations") Remark 2.1.3. It follows from Sakai's predual uniqueness theorem that when checking point-ultra weak ...
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68 views

A question on double dual of C*-algebra

Let $A, B$ be the C*-algebra. Assume $A$ is nonunital, $B$ is unital and $\phi: A \rightarrow B$ is a contractive completely positive map. Then we consider the double adjoint map $\phi^{**}: ...
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58 views

The spectrum of the operators

Let $X, Y$ be the Banach space, and $T_{1}: X\rightarrow X$ and $T_{2}: Y\rightarrow Y$ be the bounded linear operators. Then what is the relationship between $\sigma(T_{1})$, $\sigma(T_{2})$ and ...
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131 views

Arveson's Extension Theorem in C*-algebra

I am reading a book C*-algebra and finite-Dimensional Approximations. There are two conclusions (in the book) below. Corollary 1.5.16. Let $E\subset A$ be an operator subsystem and $\phi: E ...