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).
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Operators defined on pre-Hilbert spaces generated by normal semi-finite weights on a von Neumann algebra is closable
Let $M$ be a von Neumann algebra and $\phi, \psi$ be two normal semi-finite weights on $M$. Let us now set
$$
\begin{align}
&\mathcal{n}_\phi = \{x\in M: \phi(x^*x)<+\infty\},~~\mathcal{n}_\psi ...
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Is $a \mapsto(\pi(a) h \mid h)$ for $h \in \mathcal{H}$ with $\|h\|=1$ a state for non-unital C*algebra?
From the textbook Morita Equivalence and continuous trace C*algebra:
So I can show that the map $a \mapsto(\pi(a) h \mid h)$ for $h \in \mathcal{H}$ with $\|h\|=1$ is a state if our C* algebra is ...
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Does the ultra-weak topology coincide with the weak topology on the unit ball?
Just let me say first, I am no expert neither in $C^*$-algebras nor in $W^*$-algebras. But I came across the following question:
Let $A$ be a $C^*$-algebra. Then its bidual $A^{**}$ is also a $C^*$-...
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Definition of transpose map in Murphy's C* algebras
Given a continuous map $\theta : \Omega \rightarrow \Omega'$, the author defines its transpose $\theta^t : C(\Omega) \rightarrow C(\Omega')$ by $f \mapsto f\circ \theta$. Then he states that it is a ...
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Given an $*$-algebra, when I can construct a codimension 1 subalgbera?
For the case of finite type algebras, the question is answered easily by Proposition 1.1.1 in the first Kadison-Ringrose book on operator algebras, and the answer is obviously yes. Let's stick with ...
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A question about a projection in a W$^*$-algebra
Let $M$ be a W$^*$-algebra algebra over a Hilbert space $\mathcal H$. Let $N$ be a W$^*$-subalgebra of $M$ and $\phi:M \to N$ be a normal positive linear map from $M$ onto $N$. Let us consider the ...
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Solving "time evolution" partial differential eq using Lagrange shift operator
I was reading about the Weierstrass transform $(W[\cdot])$, and how it's related to the difussion equation in one dimension. It's relation is given by that $W[f]$ is the convolution with the Heat ...
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Does a subset of linear operators with trivial null space separates points?
Let $A \subset B(H)$ be a subset of bounded linear operators in a Hilbert space $H$. The null space of $A$ is defined to be the collection of all $x \in H$ such that $T(x) = 0$ for all $T \in A$.
$A$ ...
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Normal Hahn-Banach extension of a von Neumann algebra
Let $M\subseteq B(H)$ be a von Neumann algebra. Let $\omega \in M_*$ be a normal functional. In a paper I am reading$^{(\dagger)}$, it is said that it is possible to find a normal functional $\...
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Positivity of an $n \times n$ matrix in $M_n(B(\mathcal H))$
Let $\mathcal H$ be a Hilbert space and $B(\mathcal H)$ be the set of all bounded operators on $\mathcal H$. Let $B \subset B(\mathcal H)$ be a $C^*$-subalgebra of $B(\mathcal H)$. Let $\xi \in H$ be ...
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Representation of the Time Evolution Operator in QM
I was searched about the Time-evolution operator from time $t_0$ to $t$, denoted as $U(t,t_0)$ used in quantum mechanics, and which has the formula for a time-independent Hamiltonian operator $\hat H$:...
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Books for finite rank operators [closed]
I hope this message finds you well. I am currently working on a research project related to finite rank operators on Banach spaces, specifically their elementary and general properties. As part of my ...
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Any reference for the mathematics of Quantum Mechanics with infinite degrees of freedom?
I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
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Suficient condition for an unital $C^*$-algebra homomorphism to be surjective on postive elements.
I am trying to find sufficient conditions for an unital $C^*$-algebra homomorphism $\alpha:A \rightarrow B$ to be surjective on positive elements, that is, $\alpha(A_+)=\alpha(B_+)$. For example, if ...
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Murphy theorem A.1 about seminorms and continuity
I wish to prove that if a functional $\tau$ on some vector space $X$ is continuous in a locally convex topology (meaning one generated by some family of seminorms $\Gamma$) then there is a constant $M$...
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Nuclear $C^*$-algebras have the WEP-property.
Exercise: Let $A$ be a nuclear $C^*$-algebra. Show that $A$ has Lance's WEP-property, i.e. show that there exists a ucp map $\Phi: B(H_u) \to A^{**}$ such that $\Phi(a)= a$ where $A\subseteq A^{**}\...
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Approximating a self adjoint element with invertible elements
I would like to prove the following proposition:
Let $x\in \mathcal{A}$ be a self adjoint element of unital C* algebra $\mathcal{A}$. Then for any $\epsilon >0$, there is an invertible element $y\...
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A factor von Neumann algebra is a prime algebra.
Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ denotes the algebra of all bounded linear operators on $\mathcal{H}.$
Recall that a von Neumann algebra $\mathcal{U}\...
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How to show that a map is positive (positive-preserving) iff it's dual is also positive? [closed]
I was reading UTX: introduction to matrix analysis and application.
And in section 2.6 it mentioned the dual $\alpha^*$, $\alpha^*:\mathbb{M}_n\rightarrow\mathbb{M}_k$, of a linear map $\alpha$ ...
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The algebra of all bounded linear operators acting on a complex Banach space is a prime algebra.
Let $X$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B}(X)$ the algebra of all bounded linear operators on $X.$ I want to show that $\mathcal{B}(X)$ is a prime algebra.
My ...
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finite projection in a semifinite von Neumann algebra
Let $M$ be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace $\rho$. Let $p\in M$ be a nonzero projection in $M$.
If $\rho(p)=\infty$, can we deduce that $p$ is not a ...
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spectral projection of the (unbounded) self-adjoint affiliated operator
Let $M$ ba a purely infinite von Neumann algebra. Suppose that $X$ (may not be bounded) is a positive self-adjoint affiliated with $M$. We know that the spectral projection of $X$ lies in $M$.
My ...
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Do semicircular families exist?
Let $(A,\varphi)$ be a $^*$-probability space, i.e.,
$A$ is a unital $^*$-algebra, and
$\varphi$ is a $^*$-preserving unital linear functional.
Let $(c_{i,j})_{i,j\in I}$ be a positive definite ...
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Series of operators
Given an operator algebra of bounded operators $\mathcal{A}$ acting on a Hilbert space $\mathbb{H}$, I am interested in the algebra of tensor products $\mathcal{A}^{N} = \otimes_{k=1}^{N} \mathcal{A}...
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$M\xi$ forms a unital Hilbert algebra for some cyclic-separating vector $\xi$
Let $M$ be a von Neumann algebra over a Hilbert space $\mathcal H$ and $\xi\in \mathcal H$ be a cyclic-separating vector for $M$, that is, $[M\xi]=[M'\xi]=\mathcal H$, where $M'$ is the commutant of $...
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Toeplitz Operators Generate Compacts
This exercise is from Higson's 'Analytic K - Homology'.
Let $H$ be the Hardy space $H^2(S^1)$. Show that the $C^*$-subalgebra of $\mathcal{B}(H)$ (bounded operators) generated by the Toeplitz ...
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affiliated operator lies in the von neumann algebra
Let $M$ be a von Neumann algebra.
Suppose $h$ is a positive self-adjoint operator which is affiliated to the center of $M$ . If $p$ is the spectral projection of $h$， we know that $p$ belongs to the ...
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MASA in separable hyperfinite type II_1 factor [closed]
Does the separable hyperfine type $II_1$ factor contain a copy of $L_\infty([0,1])$?
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The center of a von Neumann algebra may or may not have minimal projections
Let $\mathcal H$ be a Hilbert space with $\dim \mathcal H=\infty$. Let $M \subseteq B(\mathcal H)$ be a von Neumann algebra acting on $\mathcal H$. Let $Z$ denotes the Center of $M$. Now a projection $...
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Does Hilbert $C^*$-modules have orthonormal basis?
Let $E$ be a (right) Hilbert $A$-module where $A$ is unital $C^*$-algebra. I call a subset $S$ of $E$ to orthonormal if $\langle x,x\rangle=1$ ($1$ denotes the unit in $A$) for all $x\in S$ and $\...
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Is the unit ball complete in the 2-norm of tracial von Neumann algebra
Let $V$ be a tracial von Neumann algebra i.e a von Neumann algebra with a tracial faithful normal state $\tau$. It is known that the trace induces a norm on the von Neumann algebra defined by $||a||_2^...
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Theorem 3.1.1, Gerald Murphy's C* algebra and Operator Theory
I have a doubt on the proof of the theorem. $f \in C_0(\Omega)$, and $K = \{ w \in \Omega \ | \ |f(w)| \ge \epsilon\}$ which is compact. Then by Urysohn lemma, there is a continuous function $g: \...
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Noncommutative analog compact open topology
Recall that for any locally compact Hausdorff topological space $X$, there is so called compact open topology on $C(X)$ https://en.wikipedia.org/wiki/Compact-open_topology. Is there a noncommutative ...
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Image of conditional expectation in von Neumann algebra
Let $\mathcal M$ be a finite von Neumann algebra and $\mathcal E:\mathcal M\to\mathcal N$ be the conditional expectation operator onto a von Neumann subalgebra. Suppose that we know $\|ex_ne\|_\infty\...
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Group of Integer Valued Functions on Interior of Unit Circle
This is from Higson's Analytic K-Homology.
Let $X$ be a non-empty and compact subset of $\mathbb{C}$. An index function for $X$ is an integer-valued function on the set of bounded components of the ...
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Operator topologies on direct sum of von Neumann algebras
I'm trying to understand the topological properties of infinite direct sums of von Neumann algebras (particularly in the strong operator topology).
If I take the direct sum
$$
M = \bigoplus_{\lambda \...
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Any symmetric normed ideal $\mathfrak{a}$ on $\mathcal{H}$ is linearly generated by its positive elements
I have some questions about the proof of this statement in the book "Elements of Noncommutative Geometry" by Garcia-Bondía.
A ideal $\mathfrak{a}\subset K(\mathcal{H})$ is called ...
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$\sup \{upu^*: u \in \mathcal U( M)\}$ is a central projection of a von Neumann algebra $M$
Let $M$ be a von Neumann algebra acting on the Hilbert space $\mathcal H$. Let $0 \ne p \in M$ be a projection. Now consider a projection $q \in M$ defined by
$$q:=\sup \{upu^*: u \text{ is an unitary ...
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Let $A$ be a unital $C^*$-algebra, $a\in A,\ x,y\in A_{sa}$. Does there exist a state $\phi$ on $A$ such that $\phi(xa^*ay)=\lVert a\rVert^2\phi(xy)$?
If $A$ is a commutative, unital $C^*$-algebra, then $A=C(X)$ for some compact, $T_2$ space $X$. Then $a=f,x=g$ and $y=h$ are continuous functions on $X$. Then there is $x_0\in X$ such that $\lVert f\...
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Strictly positive element in a von Neumann algebra
If $A$ is a $C^*$-algebra and $a\in A^+$. $a$ is said to be strictly positive if $aA$ is dense in $A$.
Let $M$ be a von Neumann algebra. Does there exist another sufficient and necessary condition to ...
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Residual Subgroup, Conditional Expectations, and Central Sequences
Throughout, let us assume that $G$ denotes a group with the so-called infinite conjugacy class (ICC) property; i.e., the property that that the conjugacy class of any non-trivial element is infinite. ...
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Smeared polynomials of creation and annihilation operators
Given a massive free scalar field. We can define the quantum *-algebra of observables as a subset $\mathcal{E}\subseteq C^{\infty}(\mathcal{F})$ given by
$$
\mathcal{E}:= \lbrace 1,a[f],\overline{a[g]}...
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Two projections are unitary equivalent and commute will give us self adjoint unitary equivalent?
I am reading Jones famous paper "index for subfactors" recently. And I met some questions in the reading. Here is the link for the paper:https://link.springer.com/article/10.1007/bf01389127.
...
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$C^*$-Algebras and Gelfand Duality Reference
I'm interested in learning more about the duality between (locally compact) Hausdorff spaces and commutative $C^*$-algebras. Does anyone have an introductory textbook they recommend? My background in ...
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Why is the projection to the image of an algebra in the commutator
The proof in Murphy's book (chapter on Von Neumann algebras) of the fact that a *-subalgebra $A$ of $B(\mathcal{H})$ is strongly dense in $A''$.
The proof procedes as follows:
Take a $x\in \mathcal{H}$...
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If $\varphi : A\to B$ is surjective, then is $GL_{0}\left(B\right)$ contained in $\varphi\left(GL\left(A\right)\right)$?
Let $A$ and $B$ be two $C^*$-algebras. Suppose $\varphi : A\to B$ is a surjective $*$-homomorphism. Denote the set of invertible elements of $A$ by $GL\left(A\right)$ and denote the set of invertible ...
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Von Neumann algebra decomposition as integral of factors and mixed state decomposition as sum of irreducible states
My question is the following. It is known that any Von Neumann algebra can be uniquely decomposed as integral over algebra factors. It is also know that any mixed state can be uniquely expressed as ...
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Spectrum of a hermitian element of a C* algebra is connected?
In the proof of theorem 2.1.11 from C*-Algebras and Operator Theory, by Gerald Murphy the author is trying to prove $\sigma_B(u) = \sigma_A(u)$, where $B$ is a C*-subalgebra of $A$ and $u \in B$ is a ...
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A question on disjointness preserving operator
In "Disjointness preserving operators on $C^*$ algebras" by Manfred Wolff, Arch. Math 62, 248-253, 1994 it is presented the concept of zero divisor preserving map as follows:
Let $A$...
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How to derive or understand the quantum detailed balance condition for Markov open system?
In the "On The detailed balance conditions for non-Hamiltonian systems", I learned that for a Markov open quantum system to satisfying the master equation with the Liouvillian superoperators,...
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