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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Existence of uniform multiplicity projection in abelian Von Neumann algebras.

I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...
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35 views

Polar decomposition for completely bounded linear maps

Let $M$ be a W*-algebra and $f$ be a norm one and normal functional on $M$. Polar decomposition says that, there is a unique positive linear functional, denoted by $|f|$, satisfying in: ...
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23 views

$p$-complete boundedness of homomorphisms between $L^p$ operator algebras

Let $A$ and $B$ be non-unital Banach subalgebras of $B(L^p(X,\mu))$ where $p\in[1,\infty)$. We unitize $A$ (and similarly for $B$) by considering $A^+=A+\mathbb{C}I\subset B(L^p(X,\mu))$, and we norm ...
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23 views

An operator which moves on the boundary

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$. Let ...
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20 views

Operators problem

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having ...
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24 views

A formula for representation

Let $A$ be a C*-algebra. Do you confirm the following discussion? Let us consider a representation $\pi:A\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$. If we denote ...
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26 views

Operator system of minimal dimension with one dimensional projections

Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in ...
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80 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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62 views

Weak (operator) null sequence is bounded and pointwise convergent to zero

I was reading Diestel book (Absolutely Summing Operators) and it says: "(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero." I ...
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42 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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33 views

Meaning of kernel

If something is the 'kernel' of a transformation, say $K(x,x')$, does it mean I should take the integral $$\int K(x,x') f(x')dx'$$ There are many different meanings of kernel and I did not see their ...
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32 views

Questions about stable rank of inductive limit of $C^\ast$-algebras

Let $A$ be an inductive limit of $\{A_n\}$ which are stable rank one. In Huaxin Lin's book An introduction to the classification of amenable $C^\ast$-algebra. The author assume that $\{A_n\}$ and $A$ ...
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44 views

Centralizer of $C^*$ algebra

Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra. If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for ...
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68 views

Eigenvalues and eigenvectors of certain diagonal constant matrices

Suppose I have an infinite complex diagonal constant (Toeplitz) matrix, which is also Hermitian. This is given by finite number of complex parameters $z_1, z_2, \cdots, z_k$. If, $z_1$ is the ...
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34 views

Question about Noncommutative quotients

I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are ...
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32 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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26 views

Lifting invertible elements in a $C^*$-algebra connected to the identity

Let $A$ and $B$ be unital $C^*$-algebras and suppose that there is a surjective *-homomorphism $f:A\rightarrow B$. Then any invertible element in $B$ that is connected to $1_B$ can be lifted to an ...
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47 views

Why does one only consider one-parameter groups in Borchers-Arveson theorem?

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ ...
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54 views

Commutant of algebra of multiplication operators

Let $L^2(X)$ be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space $X$. Define $\mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}$, where $M_f$ is the the ...
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35 views

Interpreting the lingo of a definition

The Terms I grew up with: A bounded linear operator $U$ on a Hilbert space $H$ is a partial isometry if there exists a subspace $M$ of $H$ such that $\|Ux\| = \|x\|$ for all $x\in M$, and $Ux = ...
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66 views

Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and ...
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92 views

Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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32 views

$C^*$ algebra generated by a $C^*$ algebra and a group

In this article, "Spectral measures in C∗-algebras of singular integral operators with shifts", in chapter 3.1. They have a $C^*$ algebra $U$, and an unitary representation $\pi$ of a discrete group ...
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50 views

Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into ...
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37 views

Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions ...
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38 views

Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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26 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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46 views

States: Approximate Unit

Given a C*-algebra $\mathcal{A}$. Consider a state: $\omega\geq0$ Especially one has: $\sup\omega(E^2)=\|\omega\|$ Can it actually fail to be a proper limit? The problem is that the square is not ...
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34 views

Prove a factor is III$_{\lambda}$ type

This question is from the Sunder's book An Invitation to von Neumann Algebras Ex 4.2.13 Let M be a semifinite factor with fns trace ${\tau}$. Let ${\theta}$ be an automorphism of M such that ...
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42 views

Wiener's lemma and Hulanicki's lemma

Let $\mathcal{A}(\mathbf{T})$ be the Banach algebra of continuous complex-valued functions on the unit circle with absolutely convergent Fourier series. Then Wiener's lemma states that if $f \in ...
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60 views

How to prove the following isomorphism?

Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow ...
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111 views

Diagonalization of total angular momentum over creation operators for an isotropic harmonic oscillator?

You have an isotropic three dimensional quantum harmonic oscillator so the Hamiltonian is $$ H=\frac{p^2}{2}+\frac{r^2}2 $$ If you do the creation-annihilation operator-algebra trick and define ...
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99 views

Bounded operators with infinite matrix representations

Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and that $I$ is a non-empty set. If $A\subseteq B(K)$ for some Hilbert space $K$, we can ...
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116 views

Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
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53 views

Where is the most clear and concise exposition of the spectral theorem for self-adjoint operators on Hilbert space?

This question is certainly subjective, which may warrant votes to close. I'm simply looking to find the "best" written exposition of the spectral theorem for possibly unbounded self-adjoint operators ...
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75 views

von Neumann Algebras and measures

I read that any abelian von Neumann algebra is isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-commutative measurable ...
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115 views

Ultra weakly continuous trace on a von Neumann Algebra

Let $M$ be a infinite dimensional von Neumann Algebra with a positive, faithful, ultra weakly continuous trace $tr:M\rightarrow \mathbb{C}$. Is it possible to show that $tr$ is strongly continuous?
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64 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
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53 views

Smoothness of distributions

I've reached an impasse in reading some texts on distribution theory, as several of them mention smooth distributions, but none of them actually define what it means. Therefore I'd like to know if ...
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27 views

A certain product of C*-algebras

So, I am looking for some kind of 'product' $\bullet$ on the category of (unital?) $C^*$-algebras satisfying that $M_n(\mathbb{C})\bullet M_m(\mathbb{C}) = M_{m+n}(\mathbb{C})$ where ...
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249 views

Cyclic Properties of the trace in Quantum Field Theory

I'm trying to figure out what's going on in this paper here between lines 10 and 11. But I'll give a brief rundown of what's going on. We are trying to compute the trace of the exponential operator ...
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19 views

Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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46 views

Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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128 views

Examples of operator theory on Hilbert space

$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$ $(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ ...
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Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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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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82 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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87 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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56 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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60 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)\\ ...