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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38 views
2 positive decomposable maps
A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
2
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1answer
54 views
Determining whether an operator is Hermitian
The operator $F$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a) $, where $a$ is a nonzero constant. Determine whether or not $F$ is a Hermitian operator.
If the condition for $F$ to be Hermitian is ...
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2answers
87 views
Recovering a group from its C*-algebras and group algebra
Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions?
Is it true that:
if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$?
if $C_r^*(G)$ and ...
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1answer
34 views
Help proving operator inequality
Given $P \geq 0$, I need to show that $2Tr(P^{5/2}) \leq Tr(P^3) + Tr(P^2)$. It's trivial to show that the RHS is the trace of a positive operator, but I'm at a loss on how to actually prove this ...
2
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2answers
51 views
Multiplicative linear functional on algebra of limit of polynomials
Let $A$ be the space of all functions which are limit of polynomials over the unit ball $D$.
Then $A$ is a commutative Banach algebra. Then how do I show that $A$ has no non zero multiplicative linear ...
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1answer
43 views
Linear functional on Banach algebra
Let $A$ be the space of all matrices of the form $\begin{pmatrix} a & b \\0 & a\end{pmatrix}$, $2\times2$ over complex field.
Then the spectrum of any element of $A$ comes out to be $\{a\}$. I ...
3
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0answers
114 views
Unitary operators - convergence problem
Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$.
Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
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46 views
Operator monotone functions
By definition, I know that a function $f$ is operator monotone if $A - B \geq 0 \Rightarrow f(A) - f(B) \geq 0$. For instance, we have $A^2 \leq B^2 \Rightarrow A \leq B$ because the root function is ...
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1answer
49 views
Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?
It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
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1answer
55 views
problem related to tensor product on Hilbert spaces
Let $K$ and $H$ be Hilbert spaces. Let $\{e_i:i\in I\}$ be an orthogonal basis of $H$. Define
$$
U_i:K\to K\overset{.}{\otimes} H: x\mapsto x\overset{.}{\otimes} e_i
$$
Assume ...
2
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1answer
95 views
Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.
If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then
$ a = u|a| $ for a unique unitary element $ u $ of $ A $.
If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $?
I ...
2
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1answer
54 views
Counterexample for a polar decomposition in von Neumann and $C^\ast$ algebras
For a von Neumann algebra, we have that partial isometry and positive operator of an operator in its polar decomposition belongs to the algebra, but in a $C^\ast$ algebra this may not be true.
Can ...
1
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1answer
53 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* ...
2
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1answer
42 views
Set of states not compact
I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course).
I think $K(H)$, the compact operators over a Hilbert space ...
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1answer
61 views
Multiplication not continuous in $B(H)$ in the strong operator topology
I'm reading this answer by t.b., and I'm only interested in the case when $X$ is an infinite-dimensional Hilbert space. Regarding Question 2, in the first bullet point he claims the following:
"Given ...
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0answers
36 views
Biduals generated by projections
This question is motivated by a similar question recently posed at MO:
http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras
In this setting, let $B$ be a Banach algebra ...
3
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1answer
51 views
Why should we use inverse homemorphism here?
I am a brand-new comer in dynamical system. I find it interesting that when defining ergodicity of classical dynamical system $(X,\sigma)$, they use $\mu(\sigma^{-1}(E))$ there. Since $\sigma$ is a ...
3
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1answer
110 views
Gelfand Topology and C*-algebras
Before we start here some notations to have no confusion:
Suppose $A$ is a commutative C*-algebra with unit. $\Sigma(A)$ is the Gelfand spectrum, given by all linear maps $\omega:A\rightarrow\Bbb{C}$ ...
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votes
1answer
100 views
When is a $*$-homomorphism between multiplier algebras strictly continuous?
The strict topology on the multiplier algebra $M(A)$ of a C*-algebra $A$ is that generated by the seminorms
$$ x\mapsto \| ax \|\qquad x\mapsto\| xa \| \qquad (x\in M(A), a\in A) $$
Whereas a ...
2
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1answer
75 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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1answer
134 views
Trace of an operator
Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive,
where trace($x$):= $\sum ...
2
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0answers
57 views
maps from a convex set to itself
Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
3
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1answer
63 views
Double centralizers in the Murphy book
I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake:
...
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1answer
52 views
Trace-preserving $\Rightarrow$ Norm-preserving?
Let $\theta\in\text{Aut}(\frak{M},\tau)$, where $\frak{M}$ a von-Neumann algebra and $\tau$ a faith, finite, normal trace. Does $\theta$ preserve the norm structure? (Here assumed, that $\frak{M}$ ...
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1answer
73 views
A reference request for sums of $C^*$-algebras
Does anyone know where I can find a reference for the following well-known fact:
Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s.
Then
...
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2answers
131 views
Composition of Fredholm Operators
If $ST$ is a Fredholm operator, then show that $T$ is Fredholm if and only if $S$ is Fredholm.
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0answers
49 views
Given $\theta(p)\neq p$ does there exist $q\leq p$, so that $\theta(q)q=0$?
Let $\mathfrak{M}$ be a vN-Alg. Let $\theta\in \text{Aut}(\mathfrak{M})$. Let $p\in\mathfrak{M}$ be a projection, so that $\theta(p)\neq p$. Is there a projection $q\in\mathfrak{M}$ with $0\neq q\leq ...
3
votes
1answer
58 views
What's the application of C*-algebra in topology?
C*-algebras are thought be be non-commutative topological spaces because of Gelfand's theorem that any commutative C*-algebra are isomorphic to C(X) for some locally compact Hausdorff space X. I've ...
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1answer
31 views
Kernel inclusion implies factorization
I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
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1answer
65 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 ...
4
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1answer
63 views
Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?
Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
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1answer
144 views
Do we have Maximal Abelian Algebras (MAAs)?
Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
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2answers
47 views
basic question on normed algebra
Prove the multiplication map $m:A\times A\rightarrow A$, sending $(x,y)\rightarrow x*y$ is jointly continuous in a normed algebra.
i can't understand what's jointly cont.? this is a problem from ...
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1answer
33 views
Skew-symmetric unitary on $\mathcal{B(H)}$
We know that there exists skew-symmetric unitary on $\mathcal{B(H)}$ when $\mathcal{H}$ is of even dimensions. In particular for $\mathcal{H}=\mathbb{C}^2$, any such matrix is scalar multiple of Pauli ...
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0answers
96 views
Spectral measures
Let $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a spectral measure on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Assume also that $E$ is compactly supported in the sense that ...
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0answers
34 views
Properties of a vector operator
Suppose I have a vector operator which angle dependence is given by $$\hat O(\theta)=A\sin\theta+B\cos\theta+C$$
What can I say about $\hat O$? Sorry, I do realize that it is a bit vague. Assume $\hat ...
3
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1answer
93 views
Compute spectral/projection-valued measures explicitly?
Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following:
...
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1answer
60 views
Is the cone of squares in a Jordan algebra a cone?
Let $A$ be a finite-dimensional Jordan algebra over $\mathbb{R}$, i.e. a finite-dimensional real vector space with a commutative bilinear product $\circ: A \times A \rightarrow A$ satisfying $(a^2 ...
2
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0answers
120 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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1answer
130 views
Inequality - tensor product, Hilbert spaces
Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{K}$ be a Hilbert space with an orthonormal basis $\{ e_i \}_{i \in I}$. Let $A$ be bounded linear operator from $\mathcal{H} \otimes \mathcal{K}$ ...
5
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0answers
82 views
K-theory for non-separable C*-algebras
Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
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0answers
49 views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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1answer
82 views
Seek results in group theory obtained by applping algebraic topology tools
After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
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0answers
57 views
Positive maps on $\mathcal{B}(\mathcal{H})$ to itself
Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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0answers
60 views
Time-derivative of an operator
Would I be right in thinking that the operator $$\hat O'(t)$$ is different from the operator $$D\hat O(t)$$ where $D={d\over dt}$, since when acting on a function $f$, the second corresponds to $$\hat ...
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0answers
89 views
Help with von Neumann algebras
Let $A$ be a von Neumann algebra and $A_*$ its predual.
a) If $\rho(a)=0$ for all $\rho\in A_*$ is it true that $a=0$?
b) Is $A_*$ a dense subset of $A^*$ (the set of bounded linear functionals on ...
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1answer
116 views
Dot products in commutators
Suppose $\hat r$ is an position operator, $\hat p$ is a momentum operator and $\vec c$ is a constant vector.
What does the commutator $[\hat p, \vec c\cdot\hat r]$ mean?
I see that you can expand ...
3
votes
1answer
107 views
Abelian von Neumann Algebras on non-separable Hilbert spaces
Is there a classification of Abelian von Neumann algebras on non-separable Hilbert spaces? For a classification of Abelian von Neumann algebras on separable Hilbert spaces, see this link.
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1answer
157 views
Maximal ideal space of $c_{\mathcal{U}}$
Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define
$$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$
which is a C*-algebra. Is there an accessible topological ...
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0answers
72 views
Commutator formula in infinite dimensions
The commutator formula states that for $A,B$ elements of a Lie algebra,
$$ \lim_{n\to \infty}\left\{ ...
