Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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10 votes
1 answer
457 views

For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?

Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$. For what kind of $C^*$ algebras $A$ does the following hold: $$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
-1 votes
1 answer
202 views

A commuting pair of isometries

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$. The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
13 votes
1 answer
982 views

Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
3 votes
1 answer
692 views

Universal $W^*$-algebras of locally compact groups: where is the error in this argument?

Let $G$ be a locally compact Hausdorff group. It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \...
0 votes
1 answer
186 views

Are the ideals in two $C^*$-algebras the same?

Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
1 vote
0 answers
103 views

Find a finite partition of unity in the centralizer of a von Neumann algebra

Let $M$ be a von Neumann algebra and $\varphi$ is a faithful normal state on $M$. Suppose that $M_{\varphi}$ is a type II$_1$ factor. Suppose we have the following conclusion: if $p$, $q$ are any two ...
6 votes
1 answer
573 views

Is the conditional expectation faithful?

Let $G$ be a locally compact group and let $H$ be an open subgroup in $G$. Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\...
4 votes
1 answer
351 views

Motivation for Heisenberg's modeling of observables

What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
4 votes
2 answers
310 views

Is the left-regular representation of a locally compact group a homeomorphism onto its image?

Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda_g f(h) = f(g^{-1}h)$, for a locally compact group. It is well-known that this is a unitary faithful and strongly-...
4 votes
1 answer
206 views

Support projection vs closed support projection of a normal state in enveloping von Neumann algebra

I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding. Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
0 votes
0 answers
84 views

Definition of an automorphism of von Neumann algebra acting on faithful normal state

$\DeclareMathOperator\Aut{Aut}$ In Theorem B, the authors define the $\Aut_{\varphi}$. I feel very confused about this definition. If $\alpha\in \Aut(M)$, when $x\in M$, it is sensible to define $\...
11 votes
2 answers
1k views

What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?

Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results: \begin{align*} (\ln D) 1 & {}= -\ln x -\gamma \\ (\ln D) x^n & {}= x^n (\psi (n+1)-\...
5 votes
0 answers
113 views

Unitary fusion category and subfactor

From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor. By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
0 votes
0 answers
134 views

Type III von Neumann algebra

Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
4 votes
1 answer
203 views

Definition of Radon measure on Takesaki's first volume

Consider the following theorem from Takesaki's first volume "Theory of operator algebras": In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...
2 votes
1 answer
192 views

Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
78 votes
3 answers
8k views

Norms of commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
13 votes
3 answers
1k views

Separable von Neumann algebra

What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...
1 vote
0 answers
83 views

irreducible subfactor inclusion and commutativity of induced projections

Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$. Let $\Omega\in H$. Does it follow that the projections onto $[N\Omega]$ and $[M'\...
1 vote
0 answers
93 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
1 vote
0 answers
116 views

On Riesz decomposition of Volterra operator

Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by $$ Tf(x) = \int_0^x f(t)\,dt.$$ Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...
8 votes
0 answers
405 views

Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
1 vote
0 answers
72 views

Spectral measure for a finite set of mutually commuting normal operators

The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway: Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
13 votes
1 answer
431 views

Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?

The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
63 votes
7 answers
4k views

What well known results with countability assumptions can be naturally extended to uncountable settings?

In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in ...
7 votes
1 answer
570 views

Are the compact and Haagerup approximation properties equivalent?

The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of Link) and the Haagerup approximation property. Let $M$ be a type ${II}_{1}$ ...
2 votes
0 answers
34 views

Matrix affine functions and matrix convex sets

In this post I'm trying to understand a small but significant part of an article by Corran Webster and Soren Winkler regarding a matrix convex generalization of the Krein-Milman theorem. (see the ...
2 votes
0 answers
86 views

Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents

The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $...
1 vote
0 answers
74 views

Tracial linear functionals on an amenable Banach algebra

This post is related to an earlier question about Kazhdan property (T). The purpose of the snippet below is to briefly summarize the background for the question in this post. Question: Does there ...
1 vote
1 answer
118 views

Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?

Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product $$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
2 votes
0 answers
55 views

Existence of a suitable smooth kernel

Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
6 votes
0 answers
96 views

Automorphisms of algebraic Clifford algebra of a Hilbert space

Let $H$ be a real separable, infinite-dimensional Hilbert space and let $$\mathrm{Cl}(H) = \mathcal{T}(H_{\mathbb{C}}) / \{v\otimes w + w\otimes w - 2\langle v, w\rangle \cdot \mathbf{1} ~|~ v, w \in ...
7 votes
0 answers
182 views

Kazhdan's property (T) for Banach algebras?

A locally compact group $G$ has Kazhdan's property (T) if the trivial representation $1_G:G\to\mathbb{C}$, $1_G(x) = 1$ for all $x\in G$, is isolated in $\hat{G}$ with the Fell topology. Bekka took ...
6 votes
1 answer
317 views

Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff 1. $A$ is a subalgebra of $B$; 2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...
7 votes
1 answer
266 views

Commutator ideal in nonunital C*-algebra

Let $A$ be a C*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$. Let $J$ denote the (not ...
5 votes
1 answer
411 views

Separable C* algebras and type I states

Let $A$ be a separable $C^*$-algebra and let $\omega$ be a state on $A$. Then there is an "orthogonal" probability measure $\mu$ on the pure state space $P(A)$ of $A$ such that $\omega(x) = \...
4 votes
1 answer
133 views

"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations

Let $H$ be a Hilbert space, which we interpret as a space of quantum states. If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
0 votes
0 answers
59 views

Are banach space representations of commutative $C^*$ algebras decomposable?

It is well known that, if $\pi:A\to \mathbb B(\mathcal H)$ is a $^*$-representation of a type I $C^*$-algebra, then $\pi$ is unitarily equivalent to a direct integral of irreducible representations. ...
8 votes
1 answer
458 views

CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra

$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras ...
4 votes
1 answer
131 views

A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
1 vote
1 answer
112 views

Continuous surjection between spectra of commutative von Neumann algebras

Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...
18 votes
7 answers
4k views

What are known examples of positive but not completely positive maps?

The only example I know of a positive map which is not completely positive is the transpose map on $M_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or ...
2 votes
1 answer
215 views

Showing a 2-by-2 matrix is a contraction

Let $S\subseteq\mathbb{T}:=\{z\in\mathbb{C}:\vert z\vert=1\}$ be a compact set such that $\operatorname{conv}S\supseteq\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\}$ and $B\in M_2(\mathbb{C})$....
8 votes
2 answers
182 views

Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras

There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
2 votes
0 answers
123 views

Why is Maycock's Brauer group of graded C*-algebras connected while Moutuou's is not?

In her thesis The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras (cf. Proposition 3.4), Ellen Maycock described the Brauer group of graded continuous trace $C^\ast$-algebras with spectrum a ...
3 votes
1 answer
121 views

Young-type inequality for bounded operator

Let $A$ and $B$ be two (non commuting) self-adjoint bounded operator acting on a Hilbert space and let $p,q>1$ such that $\frac1p+\frac1q=1$ Do we have a Young-type inequality such as $ \frac12|AB+...
9 votes
1 answer
607 views

Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"

It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
17 votes
2 answers
1k views

Connes' embedding conjecture for uncountable groups

In this topic, I will use the word uncountable group referring to groups whose cardinality is $\leq|\mathbb R|$. Notation: $R$ is the hyperfinite $II_1$-factor, $\omega$ is a free ultrafilter on the ...
27 votes
8 answers
3k views

Bimodules in geometry

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-...
3 votes
1 answer
424 views

Extension of a bounded linear functional

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with ...

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