Questions tagged [oa.operator-algebras]

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

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Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space: Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
erz's user avatar
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Limit of spectral projection of increasing sequence of positive operators

Let $\mathcal M$ be a von Neumann algebra. Suppose $(x_\alpha)\subset \mathcal M$ is bounded increasing net of positive operators converging to a positive operator $x\in\mathcal M.$ Is it true that $\...
A beginner mathmatician's user avatar
7 votes
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Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
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Corepresentations on Hilbert modules

In the seminal paper "Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres" by Baaj and Skandalis, we find the following very general definition of what a corepresentation ...
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Characterizing spectral radius using invertible elements in unital C* algebra [closed]

Consider A a unital C* algebra, I want to show that the spectral radius of a satisfies the following: $𝑟(𝑎)= $ inf$_{𝑏∈𝐼𝑛𝑣(𝐴)}||𝑏𝑎𝑏^{−1}||$, where Inv(A) is the set of invertible elements in ...
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5 votes
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Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces. Given two $\Cst$-...
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Completely positive, unital maps acting on unitary operators [solved]

Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...
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Non-separable non-postliminal $C^*$-algebra with injective $\pi\mapsto\ker\pi$

It is known that for a postliminal/GCR $C^*$-algebra the map $\pi\mapsto\ker\pi$ from (equivalence classes of) irreducible representations to their kernels is injective. If the algebra is separable, ...
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induced maps between group $C^*$-algebras

Suppose $G,H$ are two locally compact groups, if there is a injective homomorphism $\phi:G\to H$, can $\phi$ induce the $*$-homomorphism between group $C^*$-algebras $C^*(G)$ and $C^*(H)$? If it ...
math112358's user avatar
1 vote
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Topology on function spaces for pointwise convergence

Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
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Index of a particular subfactor

If a compact group $G$ acts on vN algebra factor $M\subset B(L^{2}(M))$, what would be the index of subfactor $[M^{G}:M]$? KIndly explain the answer.
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Real Rank of $M_n(A)$

The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest ...
Hannes Thiel's user avatar
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Completely positive maps and abelian C* algebra

This is a problem I encountered in Jesse Peterson's Notes on Von Neumann Algebras. I want to show the following: given C* algebra A, suppose for any C* algebra B, every positive map from B to A is ...
user151245's user avatar
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When do completely positive maps have a closed image?

Let $\mathcal{A}, \mathcal{B}$ be C*-algebras. A map $\phi \colon \mathcal{A} \rightarrow \mathcal{B}$ is completely positive (cp) if it's linear, * preserving and all of its' coordinatewise ...
Diego Martinez's user avatar
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Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$

The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas. Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
Sanae Kochiya's user avatar
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Is there a non-irreducible maximal subfactor other than two-sided TLJ?

A subfactor $N \subseteq M$ is called: irreducible if $N' \cap M = \mathbb{C}$, maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$. The two-sided ...
Sebastien Palcoux's user avatar
2 votes
1 answer
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Proof of uniqueness of predual of von Neumann algebra

I am currently reading Jesse Peterson's lecture notes on von Neumann algebras. I'm confused by lemma 4.4.2. In particular, it seems to me that Hahn Banach theorem here can only conclude the map $\phi$ ...
user151245's user avatar
13 votes
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Connes Embedding Conjecture is false [closed]

This preprint from yesterday claims to prove that Connes Embedding Conjecture fails. Since the paper is from outside Operator Algebras (Computer Science/Quantum Computing) and they actually work on ...
Martin Argerami's user avatar
7 votes
1 answer
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Morita-invertible C*-algebras

I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of C*-...
Captain Lama's user avatar
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Projection (or idempotent) graph of a $C^*$ algebra(or a ring)

In the literature, are there some research around a directed graph associated to a $C^*$ algebra or a ring $A$ whose vertices are projections or idempotents of $A$ and $e$ is connected to $f$ iff $ef=...
Ali Taghavi's user avatar
4 votes
1 answer
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Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$

Revision: According to comment of Wojowu we give a complete revise for this post. A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
Ali Taghavi's user avatar
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1 answer
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Action of hyperbolic group on von Neumann algebra

Let $G$ be a hyperbolic group. Let $M$ be a vN algebra in standard form. Can there exist a faithful action of $G$ on $M$ such that \begin{align*} \sigma_{g_n} \rightarrow I \end{align*} for some ...
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Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?

Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the ...
Zhaoting Wei's user avatar
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Removing the interior of spectrums

Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)? The motivation comes from the ...
Ali Taghavi's user avatar
2 votes
0 answers
155 views

Formula for a completely positive map

Is there a family of completely positive maps $L(A,B)$ depending continuously on two nonzero, symmetric positive semidefinite $n\times n$ matrices $A$ and $B$, such that $L(A,B)$ maps $A$ to $B$ and ...
Arnold Neumaier's user avatar
28 votes
6 answers
5k views

Any real contribution of functional analysis to quantum theory as a branch of physics?

In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
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1 answer
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Bisector Projection

Let $p,q$ be two projections of a $C^*$ algebra. A projection $l$ is called a bisector projection to $(p,q)$ if $$|pl-l|=|ql-l|$$ The motivation comes from the geometric intuition of "...
Ali Taghavi's user avatar
7 votes
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How can one define a kind of "determinant" on a reduced group $C^*$ algebra?

Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...
Ali Taghavi's user avatar
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Smooth sections of finite dimensional bundle and covering space

Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
Aleksandr Alekseev's user avatar
7 votes
3 answers
552 views

Commutant of the conjugations by unitary matrices

Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
Jochen Glueck's user avatar
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On mixing and weak mixing subalgebras of finite von Neumann algebras

Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
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On differential equation $Z'=Z^2-Z$ on a $C^*$ algebra

Let $A$ be a Banach or a $C^*$ algebra. We consider the differential equation $$(*)\;\;\;\;Z'=Z^2-Z$$ on $A$. Obviously the singularities of this systems are just the idempotents of the ...
Ali Taghavi's user avatar
1 vote
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Property gamma for type III factors

I am struggling to find a definition which uses centralizing sequences for property gamma in type III factors? If $M$ is type $\mathrm{III}$ factor, what is the exact definition of property gamma ...
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Special kind of translation and rotational invariance of the numerical range

Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true? If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\...
Piku's user avatar
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2 votes
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On crossed product of L^{P} spaces

Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true \begin{align*} L^{p}(M\rtimes G, \varphi^{M\rtimes G})\...
user136400's user avatar
2 votes
1 answer
526 views

Ultrapower of an ultrapower of von Neumann algebras

Let $M$ be a $\mathrm{II}_1$ factor, fix a ultrafilter $\omega$, we know the ultrapower $M^{\omega}$, is again $\mathrm{II}_1$ factor. The question is that what is the ultrapower of $M^{\omega}$, i.e.,...
sibani's user avatar
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5 votes
2 answers
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Unusual crossed product constructions being factors

Let $A$ be an abelian von Neumann algebra and $G$ a countable group acting on $A$. In the literature we meet usually two kinds of crossed product $A \rtimes G$ being a factor: if the action is (...
Sebastien Palcoux's user avatar
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0 answers
3k views

On prime factors

Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
sibani's user avatar
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4 votes
1 answer
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The ball formulation of the Kaplansky density theorem in nonselfadjoint algebras

Let $H$ be a separable Hilbert space. Let $A \subseteq B(H)$ be a finitely generated unital algebra. Let $M$ be the strong operator topology closure of $A.$ Let $B_A$ be the closed ball in $A$ and $...
J. E. Pascoe's user avatar
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2 votes
2 answers
182 views

Show that $(S^1)^*=B(\ell^2)$ knowing $(\ell^1)^*=l^\infty$

Is there a way to show that dual of trace class operators, $S^1$, is $B(\ell^2)$, bounded operators on $\ell^2$, knowing that dual of $\ell^1$ is $\ell^\infty$?
Ben's user avatar
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7 votes
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471 views

Abstract characterization of group von Neumann algebra (II1 factor)

The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{...
Sebastien Palcoux's user avatar
0 votes
1 answer
291 views

On conditional expectation from tensor products

Let $M$ be a $\mathrm{II}_{1}$ factor. Does there exist a conditional expectation from $M^{\otimes 2}$ to $M$ preserving the trace $\tau^{\otimes 2}$?
user136400's user avatar
3 votes
0 answers
84 views

Finite index subfactors of hyperfinite type $\mathrm{III}_{\lambda}$ factors

Let $M$ be a hyperfinite type $\mathrm{III}_\lambda$ factor. $N$ be a finite index type $\mathrm{III}$ subfactor, is it true $N$ is hyperfinite type $\mathrm{III}_{\lambda}$?
user136400's user avatar
8 votes
1 answer
249 views

Pinwheel Tilings and C* algebras, K-theory

I was reading that spaces of tilings can be related to C*-algebras and K-theory. Here is an example of the pinwheel tiling. [1] They construct a space called $\mathcal{A}\mathbb{T}_{pin}$ and show ...
john mangual's user avatar
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8 votes
1 answer
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Is the invariant subalgebra of the von Neumann algebra $L(F_k)$ isomorphic to $L(F_k)$?

Let the symmetric group $G=S_{k}$ act on the von Neumann algebra of the free group $L(F_k)$ via permuting its generators. Is the fixed point algebra under the action isomorphic to the whole algebra, i....
user136400's user avatar
1 vote
0 answers
109 views

Algebra structure on Haagerup tensor product of operator spaces

Let $A$ and $B$ be operator spaces. Is there any algebra structure on Haagerup tensor product of operator spaces such that the Haagerup tensor product becomes Banach Algebra? Any references or ideas?
Math Lover's user avatar
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0 votes
1 answer
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A subset (or subgroup) associated to a group

Edit: According to comment conversations we revise the question. Let $G$ be a group. We consider the following subset of $G$: $$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$ where $\...
Ali Taghavi's user avatar
0 votes
0 answers
127 views

Operator space tensor products

Given two Banach algebras $A$ and $B$ with operator space structure on each of them, i.e both of them are closed subspaces of $B(H_1)$ and $B(H_2)$ respectively for some Hilbert spaces $H_1,H_2$. ...
NewB's user avatar
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0 answers
107 views

On an application dominated convergence theorem in vN algebras

$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
user136400's user avatar
2 votes
0 answers
62 views

On $L^{1}(M',\tau')$

Let $M$ be a $\mathrm{II}_{1}$ factor with the faithful trace $\tau$ in standard form. Suppose $\tau'(x')=\tau(Jx'^{*}J)$. If we complete $M$ with $\|\cdot\|_{1}$ norm, i.e., $\|x\|_{1}=\tau(|x|)$, ...
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