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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3
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36 views

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 ...
1
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1answer
27 views

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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1answer
13 views

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 ...
3
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1answer
38 views

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 on 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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1answer
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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0answers
22 views

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 ...
1
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1answer
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., ...
1
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1answer
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 ...
2
votes
2answers
76 views

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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1answer
31 views

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 ...
2
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0answers
27 views

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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10 views

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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0answers
14 views

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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1answer
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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1answer
21 views

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 ...
4
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1answer
26 views

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 ...
3
votes
1answer
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 $\lVert p-q \rVert \leq 1$. ...
2
votes
2answers
43 views

$A\in B(H)$ 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, $\{a\xi | a\in A \}$ ...
2
votes
2answers
29 views

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 $...
1
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1answer
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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0answers
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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0answers
24 views

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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1answer
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 ...
1
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1answer
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 $M_2(A)^{(1)}$...
0
votes
1answer
33 views

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($y$)=...
2
votes
1answer
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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0answers
15 views

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 $S$ ...
2
votes
1answer
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 ...
0
votes
1answer
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 ...
2
votes
0answers
39 views

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 ...
2
votes
1answer
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: ...
3
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1answer
41 views

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 $A$ ...
2
votes
1answer
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 ...
2
votes
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 ...
2
votes
1answer
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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1answer
49 views

Two Fredholm operators A and B have the same index iff there is an invertible operator C s.t. A-BC is compact

First, I've shown that a Fredholm operator $T\in B(E)$, where $E$ is an infinite dimensional Banach space, is a compact perturbation of an invertible operator iff its index vanishes. Then, I need to ...
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1answer
31 views

Why is $tr((a_{ij})=\sum\limits_{i=1}^n a_{ii}$ k-positive for all $k$?

Let $A$ be a $C^*$-algebra and $$Tr:M_n(A)\to A,$$ $$(a_{ij})\mapsto \sum\limits_{i=1}^n a_{ii}.$$ The claim is that this map is k-positive for all $k\in\mathbb{N}$. Let $k\in\mathbb{N}$ and consider ...
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1answer
25 views

Quantum mechanics, operators acting on generic state

For a quantum mechanical harmonic oscillator of constant real mass m and frequency ω, define the following operators on the Hilbert space: $ h = a^{+}a $ $e = [\sqrt{-1 + a^{+}a}]a^{+} $ $ f ...
2
votes
1answer
44 views

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^...
7
votes
1answer
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 $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
4
votes
1answer
41 views

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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0answers
28 views

Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
2
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1answer
25 views

a basic question in crossed product for compact group action

I am quite new into crosssed product of Fréchet algebras or C$^*$-algebras. So if the question is too basic please excuse me. Suppose we have two Fréchet algebras or C$^*$-algebras $A$ and $B$ and ...
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0answers
14 views

Find conditions on operator $A$ , {$Af_n$} is in $L^2$ and $\lim\limits_{n\mapsto \infty} \int_a^b (Af_n - f)^2 dx =0$.

Consider a linear operator $A$ . Please could you state sufficient conditions on $A$ other than the one I gave such that for any $f$ in $L^2$ there is a sequence {$f_n$} such that the sequence {$...
2
votes
1answer
39 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
2
votes
0answers
14 views

Nonhomogeneous Toeplitz equation

Let $T$ be the Toeplitz operator on $\ell_p$ with symbol $\alpha(\lambda)=a/2\cdot \lambda-(a+1/2)+\lambda^{-1}$, where $a$ is complex. I want to solve the following $$ Tx=y $$ for $x\in \ell_p$ ...
2
votes
1answer
29 views

Tomita Theory: Involution

Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\...
3
votes
1answer
21 views

subset of pure states with norm condition already dense

I struggle to proof the following statement: Let $Y\subseteq P\left(B\right)$ a subset of pure states on a $C^*$-Algebra $B$ such that for every $b\in B$ there exists a $\varphi \in P \left(B\right)$ ...
3
votes
2answers
29 views

why is $f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$ injective?

If $f:A\to B$, $g:C\to D$ are injective $\ast$-homomorphisms between $C^*$-algebras $A, B, C, D$, is the induced map on the spatial tensor product $$f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$$...
2
votes
0answers
29 views

What is the motivation of studying $P[A]$ in operator K-theory?

I am reading the last chapter of Murphy's $C^*$-algebras and operator theory. He defines $$P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$$ and construct the Grothendieck group ...