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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190 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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38 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
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1answer
68 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
2
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1answer
32 views

The spectrum of $L:=-\Delta+V(x)$ on complex $L^2(\mathbb{R}^N)$ and real $L^2(\mathbb{R}^N)$

In general, when one talks about the spectrum of an self-adjoint operator, it is naturally considered in a complex Hilbert space (say $L^2(\mathbb{R}^N,\mathbb{C})$). Moreover, the spectral ...
2
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1answer
69 views

Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
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1answer
39 views

Isomorphy of $C_0(U)$ and an ideal

Let $X$ be a topological space, $Y\subseteq X$ closed and $U:=X\backslash Y$. Then $I_Y:=\{f \in C_o(X)\mid f_{| Y}=0\} \subseteq C_0(X)$ is a closed Ideal. I want to show that $I_Y \cong C_0(U)$. I'm ...
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1answer
70 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
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1answer
25 views

(nx(gradxn))^2 operator question?

by $A\times B \times C = (A \cdot C)B-(A \cdot B)C$, i need to expand $n \times \bigtriangledown \times n$, where all of these are vectors. Here is what i have right now $n \times \bigtriangledown ...
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1answer
70 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
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1answer
45 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 set of ...
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1answer
52 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
38 views

Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
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1answer
59 views

Soft questions: $C^\ast$-dynamical systems

I have read some papers about $C^\ast$-dynamical systems. But there are still some questions in my mind which I can not answer. When is the $C^\ast$-dynamical system introduced? Why is the ...
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1answer
41 views

positive elements in c*algebras and states

I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$. The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a ...
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1answer
24 views

limit of 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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1answer
26 views

strictly positive element iff A contains a countable approximative unit

I search a proof of: Let A be a c$^*$-algebra and let $(u_n)_{n\in\mathbb{N}}$ an approximative unit in A. Then $a=\sum\limits_{n=1}^{\infty}\frac{u_n}{2^n}$ is strictly positive. Could anybody tell ...
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1answer
59 views

Positive Elements in a C*algebra

Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with ...
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1answer
21 views

A question about Invariant subspaces of an algebra.

I feel that this is a very simple problem, but somehow I don't see the solution. I want to show that if $A$ is a subalgebra of $B(H)$ containing $1$ then if $B\in SOTcl(A)$, for every n, ...
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1answer
60 views

How do you prove $L^{\infty}$ is a C*-algebra?

If we define on $L^{\infty}$ the essential supremum norm ($\| \|_{\infty}$), then how can I prove this norm is submultiplicative ($\| T_1T_2\|_{\infty}\leq \| T_1\|_{\infty}\|T_2 \|_{\infty}\, \forall ...
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88 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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30 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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117 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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97 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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196 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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101 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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103 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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43 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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154 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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76 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 ...
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29 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 ...
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30 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 ...
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21 views

States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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28 views

An exercise about nondegenerate representation

Suppose $I\subset J$ is an inclusion of nonunital C*-algebras such that some approximate unit $\{e_{n}\}\subset I$ is also an approximate unit of $J$. If $J\subset B(H)$ be a nondegenerate ...
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49 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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33 views

Prove that there is no invertible function in the kernel of a linear functional.

STATEMENT: Use the compactness of $X$ to show that if there is no point $x_0$ such that $g(x_0) = 0$ for all $g ∈ \mathcal{N}$ , then $\mathcal{N}$ contains an invertible function. Some ...
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39 views

Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and ...
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34 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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39 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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28 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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18 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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57 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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26 views

Saturated Monotone and Increasing Mappings

Let $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a monotone mapping, i.e., $$ \left( A(x) - A(y) \right)^\top \left( x-y\right) \geq 0 $$ for all $x,y \in \mathbb{R}^n$. Let $B : \mathbb{R}^n ...
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25 views

Tensor product decomposition for algebras with non trivial center

I have a question regarding operator algebras with non-trivial center. This is with regard to defining entanglement entropy in gauge theories. Suppose there exists an algebra of operators associated ...
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52 views

Hermitian linear functional on a C* algebra is bounded

Show that every Hermitian linear functional $\phi$ on a C* algebra $A$ is bounded. A linear functional $\phi$ is said to be Hermitian if it satisfies: $\phi(x^{*})=\overline{\phi(x)}$. This is an ...
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25 views

Question about operator algebra

I'm not certain about the rules of operator algebra, and I am wondering if these statements are equivalent $$\left(z^2\frac{d}{dz}-2z\right)\cdot\left(z^2\frac{d}{dz}-2z\right)=$$ ...
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39 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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52 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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39 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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109 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 ...