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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Uncountable sequence $R_1 \subset R_2 \subset … $ of von Neumann algebras acting on separable $H$?

Can there exist an uncountable sequence $R_1 \subset R_2 ...$ of von Neumann algebras all acting on the same separable Hilbert space $H$, with a "limit" algebra $R$ such that $R_\alpha \subset R$ for ...
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All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
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Equivalence of some characterization of pure states

I'm looking for a reference or a proof for these well-known facts in $C^*$-algebras theory for which, however, I havent found any clearly written proof of the same type of ones I will sketch. ...
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32 views

A certain limit of a state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
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Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting op a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
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10 views

Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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modular operator

I could calculate a modular operator $Δ$ for a state $τ_α(x_{ij})= αx_{11}+(1-α)x_{22}(0<α<1)$. But, I cannot understand an automorphism $σ_t(x)=Δ^{it}(x)Δ^{-it}$ of $M_2(\mathbb{C})$. For ...
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34 views

Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
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A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
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22 views

Countable weighted shift has no invariant subspace.

Suppose I have $T(e_n)=w_ne_{n+1}$ where $w_n>0$ (and are bounded) and $\{e_n\}$ denotes the canonical basis of $l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ...
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68 views

If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
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38 views

Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
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Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
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Bott projection as $K_1$ class

Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book $K$-theory and $C^*$-algebras) given by $b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{...
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A comparison between absolute values of functionals

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
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56 views

Generalized Absolute Value II

Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...
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20 views

In a C*-algebra, pure states which share the same kernel are equal

I'm reading C*-Algebras by Jacques Dixmier. And in the proof of 2.9.5, it says Let $A$ be a C*-algebra. If $f$ and $f'$ are two pure states which have the same kernel, then $f=f'$. It should ...
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Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
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Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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$A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
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25 views

Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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50 views

Elementary proof that $a - 1$ is not invertible, for self-adjoint $a$ with $\lVert a \rVert = 1$

Assume $a \in A$ where $A$ is a unital $C^*$-algebra. If $\lVert a \rVert = 1$ and $a^*=a$ we know that $1 \in \sigma(a)$, the spectrum of $a$. This follows from the fact that $\lVert a \rVert = r(a) =...
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Canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$

When I read some materials in topological dynamics, I met words: "canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$" without any definition. I know that $...
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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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125 views

Direct limit of certain $C^*$ algebras is simple

Let $X$ be a compact Hausdorff space. Let $(x_n)$ be a sequence in $X$.Assume $X$ has no isolated points. Define $A_n = C(X, M_{2^n}(\mathbb{C}) )$ and define $\phi_{n+1,n} : A_n \to A_{n+1}$ by $$\...
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Do all $x$ in Hilbert space $H$ equal $U T \eta$ for some $T \in R$ and unitary $U \in R'$, when $\eta$ is separating for $R$?

Let $H$ be a separable Hilbert space on which von Neumann algebra $R$ acts; let $\eta \in H$ be a separating vector for $R$ (i.e. the zero operator is the only $T \in R$ such that $T \eta = 0$); let $...
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Topology whose restriction to some sub-von-Neumann-algebra is its WOT?

Let $R \subset S$ be distinct von Neumann algebras having a separating vector in the separable Hilbert space $H$ on which they act. In what cases (if any) does there exist a topology $\tau$ on $S$ ...
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17 views

the standard even grading on $M_2(A)$ and $A\otimes \mathbb{K}$

I have a question about a passage in Blackadar's book about K-Theory. Let $A$ be a (ungraded) $C^*$-algebra. There is a grading on $M_2(A)$ with $M_2(A)^{(0)}$ the diagonal matrices and $M_2(A)^{(1)}$...
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If a positive operator $y$ has the same kernel as $cy$, what can we conclude about the kernel of $c$?

Let us consider the equation $x=cy$ in $B(H)$. Assume that: $y$ is a positive operator. $x$ and $y$ have the same null space. Ker($y$) is contained in Ker($c$). Can we conclude that Ker($y$)=...
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51 views

Let $N$ be a normal operator. When $W^*(N)$ is a MASA?

I am trying to find conditions in which $W^*(N)$ is a MASA, where $N$ is a normal operator acting on some Hilbert space. I know that the multplication algebra $\{M_f| f\in L^{\infty}(X, \mu) \}$ is ...
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41 views

A factorization for operators

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$. Q) Can we ...
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46 views

If $\|p-q\|<1$ then they are Murray–von Neumann equivalent- proof

$\newcommand{\ran}{\operatorname{ran}}$ Let $p$, $q$ be projections in a $C^*$-algebra $A$. If $\|p−q\|<1$ then $p$ and $q$ are MvN equivalent. I'm tryind to understand the following proof: ...
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Why are functional calculi interesting?

In my class on spectral theory we have defined the continuous functional calculus for normal elements of a C*-algebra. We were told that this is one of the most important results in spectral theory on ...
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52 views

A criterion for vector states to be in the same irreducible representation

A little wish...: is there a theorem that corresponds or implies the following Let $A$ be a $C^*$ algebra with the data of a representation in $B(H)$. Let $x,y$ be two vectors and call $S(x,y)$ the ...
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Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra (...
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projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge Q^...
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non abelian von Neumann algebras

I'm not familiar with von Neumann algebras, but I need the following fact (if it's true) for an other proof. Let $H$ be a Hilbert space, $A\subseteq L(H)$ a non abelian von Neumann algebra. Must $A$ ...
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Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
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Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continuous normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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29 views

Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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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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Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...
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1answer
46 views

What does a homomorphism $\phi: M_k \to M_n$ look like?

Let $\phi : M_k(\Bbb{C}) \to M_n(\Bbb{C})$ be a homomorphism of $C^*$-algebras. We know that $\phi$ decomposes as a direct sum of irreducible representations, each of them equivalent to the identity ...
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23 views

GNS-Construction: Involution

Given a C*-algebra $\mathcal{A}$. (It may or may not contain identity!) Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$ Construct its ...
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if $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$?

Suppose that $A_n,A$ are self-adjoint operators in $B(H)$. If $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$? From Proposition. 2.3.2 of Pederson'book, I know the result holds in ...
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37 views

Sequences of operators in $B(H)$ which converge in the weak operator topology

Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has ‎$$‎‎‎‎\...
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82 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
2
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1answer
56 views

Question about special $C^*$-algebra

I have a question about a $C^*$ algebra $A$ namely $M_2(\mathbb{C})$. I want to prove that every state $\alpha$ of $M_2(\mathbb{C})$ (thus a positive linear functional with norm $1$) is of the form $\...