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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5
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1answer
254 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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1answer
50 views

Lifting a unitary to a partial isometry

What is an example of a unital $C^*$-algebra $A$ and an ideal $I$ such that some unitary element in $A/I$ cannot be lifted to a partial isometry in $A$? Or can it be shown using general properties of ...
2
votes
1answer
79 views

Morita equivalence and KK-theory

Let $A,B,C$ be $C^\ast$-algebras. Suppose $B$ and $C$ to be strongly morita equivalent. Then $KK(A,B)\cong KK(A,C)$. Could someone provide a reference or proof of this fact? I guess the ...
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79 views

Lorentz group and eigenvalues

For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat ...
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55 views

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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58 views

Arveson index of a completely positive map on matrix algebra

Can someone tell me what is Arveson index of a completely positive map. What I want is given a map \begin{eqnarray} \psi:\mathcal{B}(\mathbb{C}^m)&\longrightarrow&\mathcal{B}(\mathbb{C}^n)\\ ...
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0answers
58 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
1
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0answers
125 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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0answers
273 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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0answers
39 views

A map that is $(n-1)$-positive but not $n$-positive

Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
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77 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 ...
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0answers
45 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 ...
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0answers
73 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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0answers
68 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 ...
1
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0answers
114 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
96 views

Definition by commutation property on structures : continuity and where?

(This is very vague, so sorry if there are approximations) I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a ...
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0answers
32 views

A looser condition for these operators?

Let $q_i,p_j$ be canonically conjugate operators for $i,j=1,2,\ldots,n$ that satisfy the relation $[p_i,q_j]=c\delta_{ij}$ where $c$ is constant and $[\cdot,\cdot]$ is a commutator. What sort of ...
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0answers
126 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
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0answers
103 views

When the ultrastrong closure of a *-algebra contains the double commutant

As lemma 6 on p.44 of Dixmier's book on Von Neumann algebras, he states that if $A$ is a *-algebra (i.e. possibly without identity, not necessarily closed in any topology) of operators in $B(H)$ such ...
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0answers
201 views

Convergence of net sums of complex numbers, as well as operators

I have some questions concerning convergence of sums where the summands are complex number, although the real motivation of my question comes from Von Neumann algebras where sometimes the summands are ...
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0answers
103 views

Affine Homeomorphism between a compact set K and the state space on A(K)

Let $V$ be a locally convex space, and let $K$ be compact set in $V$. Define $A(K)\subset C(K)$ as $A(K)=\{\phi:K\rightarrow \mathbb{C}\; |\; \phi\; \text{is continuous and affine}\}$. Then we know ...
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109 views

Extreme points and Matrix Extreme Points

With reference to this paper. Let $V$ be a locally convex space, and $K=(K_n)$ be a compact matrix convex set in $V$. Then as proved in Cor 3.6 in the above paper, we see that if $v\in K_n$ is a ...
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0answers
45 views

Does the locality or non-locality of operators imply matrix structure?

I understand that an operator, $\hat{O}$, is said to be non-local if $$b(x)=\hat{O}a(x)=\int dx'O(x,x')a(x')$$ that is, to find $b(x)$ at aparticular value of $x$, we need to know ...
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0answers
34 views

Find the probability that a measurement results in a certain interval

I was given this problem: Let the state $f(x)=e^{-|x|}$ and an operator $P=-i\frac{d}{dx}$. (a) What is the probability that a measure of P results in the interval $[0,1]$ (b) What is the Fourier ...
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0answers
156 views

A form of the Baker-Hausdorff equation

I wonder how many different ways are there of writing the Baker-Hausdorff equation! This is a form which I recently encountered and haven't been able to figure out how it comes, $e^ae^Xe^b = ...
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0answers
26 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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0answers
13 views

Reference request for various proofs of Bott periodicity for operator algebra K-theory

I am aware that there are multiple proofs of Bott periodicity for operator algebra K-theory and I would like to ask for references for them. The approach modeled after Atiyah's from topology can be ...
0
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0answers
27 views

“+” operator placed as index

What is the meaning of $(a-b)_+$? In other words, what is meant by the "+" operator when it is placed as an index. If I am comparing for example two variables $a$ and $b$. So what is the value of ...
0
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0answers
15 views

How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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0answers
32 views

Gelfand representation of the algebra $C^1([0,1])$

In Murphy's book about C* algebras, exercise question 1.10 asks the reader to show that the Gelfand representation of the algebra $C^1([0,1])$ is not surjective. Just before the reader is asked to ...
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0answers
19 views

C* Algebra, f(x,z)

Let $A$ be a $C^*$ algebra, $x\in A$ and $||x|| < 1$. Let $f(x,z) = (1-x x^*)^{-\frac{1}{2}}(1+zx)$, $|z|=1, z\in \mathbb{C}$, $\mathbb{C}$ is the complex field. How to prove: $$ f(x,z)^* f(x,z) ...
0
votes
0answers
8 views

Determining spectral bounds variationally.

I'm learning C0-semigroup theory (mainly from Arendt et al. (vector-valued Laplace-transforms and Cauchy problems), Engel & Nagel (One par. semigroups for linear evolution eq.),Evans (partial ...
0
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0answers
16 views

States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
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0answers
23 views

Is $\|a^2\|≤\|\Gamma(a)\|\|a\|$right?

For any $a$ in a commutative Banach Algebra with Gelfand representation $\Gamma$, is this inequality $\|a^2\|≤\|\Gamma(a)\|*\|a\|$right? If so, how to prove it? I'm exhausted for many hours' attempt ...
0
votes
0answers
37 views

Existence of certain subalgebras of $C(X)$

Suppose $A$ is a commutative Banach algebra.Knowing that the Gelfand transform is not surjective but injective does it imply that $A$ is not isomorphic to $C(X)$ ? by $M$ we mean the maximal ideal ...
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0answers
90 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
0
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0answers
233 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
0
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0answers
34 views

Spectral decomposition of a an Operator on $\mathcal{L}(l^2(\mathbb{N})$

I have an exercise, where I do not really know how to solve it. Let $\{\lambda_n\}_n$ be a counting of the rational numbers in $[0,1]$ and define a bounded self-adjoint operator $T$ on ...
0
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0answers
50 views

Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
0
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0answers
60 views

A isomorphism between full group C*-algebras of free group

Fix $n\in \mathbb{N}$ and let $\mathbb{F}_{n}$ be the rank-$n$ free group, can we use the universal property to illustrate the following isomorphism: $$C^{*}(\mathbb{F}_{n}\times \mathbb{F}_{n})\cong ...
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0answers
39 views

Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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0answers
47 views

Matix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
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0answers
32 views

Scalar Products on the Rational Function Field

Let $\mathbb{R}(t)$ be the rational function fields over $\mathbb{R}$. Are there scalar products $\langle -,- \rangle$ on $\mathbb{R}(t)$ such that multiplication with $t$ is selfadjoint, i.e. ...
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0answers
20 views

About Antilinear (possibly Unbounded) Operators

Let $T$ be an unbounded anti-linear operator on a Hilbert Space. I would like to know if there is a natural or easy way to see existence of adjoint of $T$, closability of $T$(such as when $T^*$ is ...
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0answers
69 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
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82 views

Abelian group C*-algebras

Let G is a locally compact Abelian group $C^*$-algebra, then $C^*(G)$ is an Abelian $C^*$-algebra, so C*(G) is isomorpohism with the C$_0$(X) for some locally compact Hausdorff space X, here X is the ...
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0answers
41 views

A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...
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53 views

self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
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42 views

Intuition behind a braid operator which is also a solution for Yang-Baxter equation

I am going through this paper, 'Quantum entanglement and topological entanglement' by Louis H Kauffman and Samuel J Lomonaco Jr published in New Journal of Physics 4 (2002). It started with ...
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116 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 ...