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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Spectrum bilateral shift

Let $U \in \mathbb{B}(\ell^2(\mathbb{Z}))$ be the bilateral shift. I want to shoow that $\sigma(U)=\mathbb{T}$. Using functional Calculus I have shown that $\sigma(U)\subseteq\mathbb{T}$. In order to ...
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Unitarily equivalent $C^*$-algebra representations

the situation i want to talk about is the following: $(H_1,\varphi_1),(H_2,\varphi_2)$ irreducible representation of a $C^*$-algebra $A$. A bounded operator $T:H_1\rightarrow H_2$ such that ...
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What are non-tagential limits?

I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions ...
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132 views

application of c*algebras

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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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 ...
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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 ...
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63 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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ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
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About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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(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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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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A separating set which is not cyclic

Let $H=L^2[0,1]$ , $T_g$ be the multiplication operator on $H$, i.e. $f\to fg$ . Let $A$ be the set of the $T_g$ as $g$ runs through the set of polynomials with complex coefficients. Let $h$ be te ...
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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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47 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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19 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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49 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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How do I compute the specific map between two isomorphic finite C* algebras?

Starting with a finite C* algebra $\mathcal{A} \subset M_{n}\left({\mathbb C}\right)$ (complex $n\times n$ matrices), $\mathcal{A}$ is known to be isomorphic to a canonical algebra of the form ...
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Interchange exponential of operators in quantum mechanics

What is the formula for interchanging products of exponential operators in quantum mechanics., i.e. I want to write $e^Ae^B = e^{B+...}e^A$
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Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
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83 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 ...
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a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
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Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
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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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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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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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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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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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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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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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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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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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Spectral radius as the inf of norms of conjugates

I need help with the following problem: Let $A$ be a unital $C^{*}$-algebra. (a) If $r(a)<1$ and $b=(\sum_{n=0}^{\infty}a^{*n}a^{n})^{1/2}$, show that $b\geq 1$ and $||bab^{-1}||<1$. (b) ...
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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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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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99 views

Bounded Operator with Closed Range

I've read Martin Argerami's answer to this question. On the first line he claims that the range of $T$ is closed. Can somebody explain me why that's the case? For me it is not necessarily closed.
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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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polars in functional analysis in Dixmier

On page 39 of Dixmier's text on Von Neumann Algebras, he argues for Lemma 1, in which he tries to see that $\theta(L_1)=E_1$ using an argument about polars from functional analysis. I was hoping ...