Skip to main content

Questions tagged [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

Filter by
Sorted by
Tagged with
2 votes
1 answer
54 views

Are there free ternary ring of operators?

I am interested in separable ternary rings of operators. For separable $C^*$-algebras we have the maximal group C*-algebra of the free group on countably many generators that quotients onto every ...
Tomasz Kania's user avatar
  • 16.5k
0 votes
0 answers
28 views

Find a partial isometry in a von Neumann algebra [closed]

Let $M$ be any factor and $\rho$ be a faithful normal state on $M$ such that $M_\rho$ is a type II$_1$ factor. Suppose that $c$ lies in the point spectrum of the modular operator $\Delta_\rho$. Can we ...
mathbeginner's user avatar
  • 1,865
1 vote
1 answer
60 views

Proving an Inequality in Hilbert Space: $\int_0^1 \chi_{[t,\infty)}(T) dt \le T$ for $T\ge 0$

Let $\mathcal H$ be a Hilbert space and $\mathcal B(\mathcal H)$ denotes the set of all bounded operators on $\mathcal H$. An element $T\in \mathcal B(\mathcal H)$ is positive, we write $T\ge 0$ if $\...
DenOfZero's user avatar
  • 127
-1 votes
1 answer
28 views

Intersection of a set with a von Neumann algebra

Let $A$ and $B$ be von Neumann algebras. Denote $W^*(A\cup B)$ by the von Neumann algebra generated by $A\cup B$. Suppose that there exists a set $C$ such that $C\cap A=\{0\}$ and $C\cap B=\{0\} $. ...
mathbeginner's user avatar
  • 1,865
0 votes
0 answers
22 views

Why is the von Neumann inequality not always fulfilled for n-tuples of commuting contractions? Why cant we just take the single dilations and get it?

Let $(T_1, \ldots, T_n)$ be an $n$-tuple of contractions in a Hilbert space $H$. We know, that for every single $T_i$, there exists a unitary operator $U_i$ in a Hilbert space $K_i$, such that $$T_i = ...
S-F's user avatar
  • 41
2 votes
1 answer
44 views

Motivation for primitive ideals of a C*-algebra

I have recently learned about the primitive ideals and prime spectrum of a C*-algebra. I am looking for a 'reason' for why they are useful. I mean this in the sense that if I was a mathematician ...
blomp's user avatar
  • 591
-1 votes
0 answers
39 views

some questions relating to comutant of von Neumann algebra s

Let $A$ and $B$ be von Neumann algebras. Set $C:=A'\cap B$. Can we have $C'=W^*(A\cup B')$? If $A$ is a type III factor,we know that $A'$ is a type III factor. Suppose that $A$ is a type III$_1$ ...
mathbeginner's user avatar
  • 1,865
1 vote
1 answer
38 views

Approximate identity and positive element condition

Let $u_\lambda$ be an approximate identity of a $C^*$-algebra $A$. If $A$ has an identity $I$, a nonzero selfadjoint element $a$ is positive if and only if \begin{equation} \left\|I - \frac{a}{\|a\|}\...
Salangidae's user avatar
2 votes
0 answers
45 views

Does the trace of an operator commute with time derivatives of an operator?

I want to find the rate of entropy production in a quantum system using von Neumann entropy $$S = -tr{(\rho \ln{\rho})}$$ by taking it's time derivative. Can I take the derivative inside the trace or ...
wednesdaypotter's user avatar
1 vote
1 answer
22 views

Projection in a hereditary subalgebra of a purely infinite C*-algebra

Let $A$ be a simple non-zero purely infinite C*-algebra. Let $p\in A$ be a projection. Then $E=pAp$ is a hereditary subalgebra. Since $A$ is purely infinite, $E$ has an infinite projection $q$. Q. Is $...
Panini's user avatar
  • 337
2 votes
1 answer
112 views

Bishop's approximation theorem

I am trying to study the generalization that Axler made to Cuckovic work on the commutants of $T_{z^n}$ (Toeplitz operator with symbol $z^n$) on $L^{2}(\mathbb{D},dA)$, in the resource I am using, the ...
euleroid's user avatar
3 votes
1 answer
47 views

How to extend the slice map $S_f: A\otimes B \to A$ given $f \in B^*$ to the multiplier algebras?

I have two questions regarding the slice map $S_f: A\otimes B \to A$ for $f \in B^*$ defined by $S_f(\sum a_i\otimes b_i)= \sum a_i f(b_i)$. (1) What is the canonical way of extending this map to the ...
Squidgame's user avatar
1 vote
1 answer
39 views

diffuseness of a corner of a von Neumann algebra

Let $M$ be a finite von Neumann algebra, $B$ a von Neumann subalgebra of $M$, and $p$ be a projection in $B$. Given that $B'\cap M$ is diffuse, is it true that the relative commutant in the corner $(...
HighwayStar's user avatar
1 vote
1 answer
114 views

Operator inner product

Let $\mathcal S(H)$ be the set of linear self-adjoint operators on Hilbert space $H$ with inner product $\langle\cdot,\cdot\rangle$. For $A, B\in \mathcal S(H), v=\sum_{i,j}a_{i,j}u_i\otimes u_j\in H\...
Hans's user avatar
  • 9,927
0 votes
0 answers
44 views

Modular conjugation with respect to a normal faithful state

According to Tomita's theorem,we have $JMJ=M'$. I wonder the relationship between $JM_{\omega}J$ and $M_{\omega}'$, where $\omega$ is a faithful normal state on $M$. Is it possible that $JM_{\omega}J=...
mathbeginner's user avatar
  • 1,865
2 votes
1 answer
51 views

Are the Hermitian linear functionals of a $C^{\ast}$-algebra necessarily bounded?

Let $\mathcal{A}$ be a (complex) $C^{\ast}$-algebra, and let $\varphi$ be a linear functional of $\mathcal{A}$. $\varphi$ is said to be hermitian if and only if $$\forall x \in \mathcal{A}\text{,}\...
Emilio Mora's user avatar
1 vote
1 answer
44 views

Taking a supremum over positive elements in the center of a von Neumann algebra

Let $(M,\tau)$ be a finite tracial von Neumann algebra with center $Z$ and standardly represented on $L^{2}(M)$. Let $z_{0}\in Z^{+}$. I want to show that $\|z_{0}\|_{\infty}^{2}=\text{sup}\{\tau(z_{0}...
SihOASHoihd's user avatar
  • 1,926
2 votes
2 answers
80 views

Appendix to Finite von Neumann Algebras and Masas

In Section A.3 of Finite von Neumann Algebras and Masas the authors write "Throughout this section, let $M$ denote a finite von Neumann algebra with centre $Z$ and centre-valued trace $\mathbb{T}$...
SihOASHoihd's user avatar
  • 1,926
0 votes
0 answers
36 views

What does it mean for an algebra to be block diagonal?

Trying to understand the proof of the following: Let $\Phi: B(H) \to B(H)$ be a unital quantum channel with Kraus operators $A_i$ for $i = 1, \dotsc, n$. Then the algebra generated by $A_1, \dotsc A_n$...
Mara Jade's user avatar
4 votes
0 answers
48 views

Higson's homotopy invariance result

I am learning about operator algebras and $KK$-theory, a result I find very striking is the following : Any split-exact $K$-stable functor $F : C^*\text{-alg} \to \text{Ab}$ is necessarily homotopy ...
Thil's user avatar
  • 81
0 votes
2 answers
73 views

Show that every finite-dimensional $C^*$-algebra can be faithfully represented on a finite-dimensional Hilbert space

Show that every finite-dimensional $C^*$-algebra can be faithfully represented on a finite-dimensional Hilbert space. [Hint: Show that finitely many states suffice for the Gelfand-Naimark theorem.] ...
francisco pires's user avatar
6 votes
1 answer
118 views

Question about continuous functional calculus and its application

I recently started learning about the topic functional calculus. My problem is that I have no idea on how to use it for, say, solving problems, exercises etc. Here is a short review of what I learned ...
Philip's user avatar
  • 635
1 vote
1 answer
79 views

A circular argument in the proof of Proposition 2.3.1 of Bratteli-Robinson

I think the proof of Proposition 2.3.1 of Operator Algebras and Quantum Statistical Mechanics 1 by Bratteli and Robinson, contains a circular argument. The book uses the following inequality \begin{...
Salangidae's user avatar
1 vote
1 answer
29 views

Multiplicative linear functional on noncommutative unital Banach algebra is bounded and norm-decreasing

It is a standard theorem that, for an abelian unital Banach algebra, every nonzero multiplicative linear functional is bounded and has norm at most $1$. I don't see why the proof can't work word-by-...
user760's user avatar
  • 1,670
0 votes
0 answers
40 views

A Question about the Matrix Representaition of an Operator

I'm reading an argument which assumes that $H$ is a Hilbert space and $A\in B(H)$.I'm having difficulty understanding one matrix representaion of $A$. First, the author decomposes $H$ as $H=\overline{...
OSCAR's user avatar
  • 573
0 votes
1 answer
75 views

What is the canonical action of a groupoid on its unit space?

Supposedly, a groupoid $G$ acts canonically on its unit space $G^{(0)}$. What is this action explicitly? I think this is the action where an arrow takes its source unit element to the target/range of ...
Panini's user avatar
  • 337
0 votes
1 answer
82 views

Commutator Ideal of Toeplitz Operators

Suppose $H^{2}$ is a Hardy space and $T_{\phi}$ is Toeplitz operator on $H^{2}$ with symbol $\phi\in L^{\infty}(\mathbb{T})$. $\mathcal{A}$ is a C*-algebra generated by $\{T_{\phi}:\phi\in L^{\infty}(\...
Halmos's user avatar
  • 23
0 votes
0 answers
39 views

For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous?

For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous? Where $A\times B$ has product topology and $A\otimes_{\min} B$ the norm topology. If yes, a proof (or reference) ...
kreitz's user avatar
  • 31
0 votes
0 answers
50 views

$C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
phy_math's user avatar
  • 6,490
0 votes
1 answer
57 views

Question about hyperstandard von Neumann algebras and selfpolar cones

Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido, second edition: I have two questions: (1) Why is $\mathscr{M}_q= q\mathscr{M}q \...
Andromeda's user avatar
  • 840
0 votes
1 answer
40 views

A positive operator $A$ satisfying $\sigma(A)\setminus\{0\}=\{b\}$ is a multiple of some projection

I'm reading an argument which includes the claim that if $H$ is a Hilbert space and $|A|$ is the absolute value of some $A\in B(H)$ and has exactly one non-zero element in the spectrum (i.e.,$\sigma(|...
OSCAR's user avatar
  • 573
0 votes
1 answer
74 views

If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.

Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem. First, the author decomposes ...
OSCAR's user avatar
  • 573
1 vote
1 answer
39 views

Is the cut-down of a positive invertible element postive and invertible?

Let $\mathfrak A$ be a unital C*-algebra; let $A \in \mathfrak A$ be such that $\sigma_{\mathfrak A}(A) \subset [a,\infty)$ for some $a > 0$; let $p \in \mathfrak A$ be a (self-adjoint) projection. ...
MakeOperatorAlgebrasGreatAgain's user avatar
0 votes
0 answers
44 views

Does $f(xy)=f(x)f(y)$ for a continuous function $f$ and normal elements $x,y$ in a C*-algebra?

In words, does a function in the continuous functional calculus behave like a homomorphism? I know that $f^n(x)=f(x)^n$ but that does not help here.
Panini's user avatar
  • 337
2 votes
1 answer
54 views

Question about the support of a normal weight on a von Neumann algebra

Consider the following fragment from Stratila's book "Modular theory in operator algebras": I understand everything in this fragment, except that equation $(3)$ holds. How can one show this ...
Andromeda's user avatar
  • 840
2 votes
1 answer
40 views

$s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
Andromeda's user avatar
  • 840
1 vote
1 answer
30 views

states and density matrices, compactness

I just started reading about the $C^*$-algebras so I'm hoping that what I'm writing won't be awfully wrong. I had a doubt regarding the connection between states in an algebra and density operators in ...
ana's user avatar
  • 75
1 vote
1 answer
57 views

Making sense of certain vector-valued integrals in von Neumann algebra theory.

Let $M\subseteq B(H)$ be a von Neumann algebra and $\varphi$ a weight on $M$ with modular automorphism group $\{\sigma_t\}_{t\in \mathbb{R}}$. We define $M_\infty$ to be the set of all elements $m\in ...
Andromeda's user avatar
  • 840
2 votes
0 answers
45 views

Normality of the Lebesgue integral with respect to a Haar measure

Let $G$ be a locally compact second countable topological group and let $\mu$ be a Haar measure on $G$. I want to show that the weight $\varphi : {L^\infty(G)}^+ \to \overline{\mathbb R_+}$, $f \...
Valentin Massicot's user avatar
3 votes
1 answer
61 views

Tensor product and extension of $\sigma$-weakly continuous linear map.

Let $M$ be a Von Neumann algebra and let $\Delta$ be a $\sigma$-weakly continuous unital $*$-morphism. We say that $\Delta$ is a comultiplication if $\Delta$ satisfies $(\Delta \otimes \iota)\Delta = (...
Valentin Massicot's user avatar
1 vote
1 answer
65 views

Finding Annihilation Operator in Space of Hermite Polynomials

This inquiry is related to my previously asked question entitled 'Proof of the Orthogonality of Hermite Polynomials' upon which I have defined a certain space with the following basis, $$ \psi_n(x) = [...
Hooman Puyandeh's user avatar
5 votes
2 answers
122 views

Normal character on a group von Neumann algebra

For a locally compact group $G$, I will denote by $L(G)$ its group von Neumann algebra, which is the von Neumann algebra acting on $L^2(G)$, generated by the image of left regular representation $\...
Mogget's user avatar
  • 641
2 votes
1 answer
33 views

Existence of uniformly convergent sequence of polynomials converging to an analytic function

Let $\Delta_1,\Delta_2,\dotsc,\Delta_n$ be mutually disjoint open discs in the complex plane. Question part 1: Does there exists a sequence of polynomials $(p_n)_{n\in \mathbb N}$ converging uniformly ...
MakeOperatorAlgebrasGreatAgain's user avatar
1 vote
1 answer
26 views

Doubt on exercise on Von Neumann factor

I am trying to solve the last point of the following exercise, but I don't know how to approach it.The exercise says: Let $M \subseteq B(\mathbb{H})$ be a type $\mathbb{II}_1$ factor Von Neumann ...
MBlrd's user avatar
  • 199
1 vote
1 answer
28 views

Nonunital weak operator closed self adjoit subalgebra has a projection that acts like an identity operator?

Let $A$ be weak operator closed self adjoint subalgebra of the bounded operators for a Hilbert space $H$(may not have identity so cannot use usual von neumann alegbra properties). I have shown that if ...
3j iwiojr3's user avatar
1 vote
2 answers
58 views

double commutant of unitary operators of a Von Neumann algebra.

Let $M$ be a Von Neumann algebra. I am having trouble showing that the double commutant of unitary operators for $M$ is equal to $M$. I have shown it with projections using borel functioncal calculus. ...
3j iwiojr3's user avatar
0 votes
1 answer
31 views

Commutator on a finite-dimensional $\mathcal{H}$ is scalar multiple of an identity

Suppose $A$ and $B$ are operators on a finite-dimensional Hilbert space and suppose that $[A, B] = c I$ for some constant c. Show that $c = 0$. I have tried approaching the problem using matrices $A, ...
Tomáš Macháček's user avatar
0 votes
0 answers
31 views

strong operator functional equvialent to weak operator continuous. A bit confused about a specific part in the proof.

I never really completely understood the first part of the proof for (iii) implies (i) even though I understood the rest of it. The thing that confuses me is that I know that a function $f:X \...
3j iwiojr3's user avatar
3 votes
1 answer
56 views

Operator on reduced group $C^*$-algebra induces operator on von Neumann algebra

Let $\Gamma$ be a discrete group. Consider its reduced group $C^* $-algebra $C_\lambda^* (\Gamma)$ and von Neumann algebra $L(\Gamma) = C_\lambda^* (\Gamma)'' \subseteq B(\ell^2(\Gamma))$. Let $T:C_\...
Tomás Pacheco's user avatar
0 votes
0 answers
28 views

Sum of bounded linear operators on a Hilbert space is bounded

I am trying to prove that $B(\mathcal{H})$ is closed under operator addition using only the notion of boundedness that we have for topological vector spaces, and I am running into difficulties. I ...
lanf's user avatar
  • 424

1
2 3 4 5
72