Questions tagged [oa.operator-algebras]
Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
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Let $(N_1,N_2)$ be commuting normal operators. Does an operator $V$ exist such that $N_i = V^{-1}T_iV$ for $(T_1,T_2)$ being contractions and $i=1,2$?
My PhD advisor and I are trying to prove something, and in our case, we would need to show, that two commuting normal operators $N_1, N_2 \in B(H)$ are similar to two commuting contractions $T_1, T_2 \...
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Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras
Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)?
Such groups must necessarily be non-...
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1
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Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
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Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?
Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
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A subfactor of finite index
We say $N$ is a subfactor of finite index of the factor $M$ if there is a normal faithful conditional expectation of $M$ onto $N$ that has finite index.
If $N$ is a subfactor of finite index of the ...
- 377
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Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
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Automorphism-invariant positive linear functionals on $C*$-algebras
Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
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positive invertible maps which are not *-automorphisms
Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
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Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
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type II$_1$ subalgebra of type III$_1$ factor
Let $M$ be a type III$_1$ factor and $N$ be the type II$_1$ subalgebra of $M$.
What is the type of $N'\cap M$? Can it be any type?
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Infinite tensor product of Hilbert spaces [duplicate]
Recently while reading an article I came across the usage of infinite tensor product of Hilbert spaces. I have got a basic understanding of doing computations in infinite tensor product while reading ...
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Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
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Pairwise orthogonality for partitions of unity in a *-algebra
Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
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Is there a notion of point in noncommutative geometry?
It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
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What are alternative or equivalent definitions of a positive-definite function on a group?
The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
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Reconstructing the manifold from space of functions in quantum mechanics
Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$.
But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the ...
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Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
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Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
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Unitary versus isometric operators
Let $\mathbb H$ be a Hilbert space, and let $\mathcal B(\mathbb H)$ be the space of bounded operators on $\mathbb H$, equipped with the operator-norm topology. Let
$\mathbb R\ni t\mapsto A(t)\in \...
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Are umbral moonshine and umbral calculus connected?
In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
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Question about the ergodic mean
This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...
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Nontrivial examples of locally compact quantum groups
What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
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What is meant by saying that the Shilov boundary of the polydisc $\mathbb D^n$ is $\mathbb T^n\ $?
Let $A$ be a complex Banach algebra and $M_A$ be the space of all non-zero multiplicative linear functionals on $A$ equipped with the weak$^*$-topology. Let $\widehat A$ be the image of $A$ under the ...
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A certainty principle?
Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where
$$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
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Approximately semifinite factors
For the sake of this question, lets call a factor $M$ approximately semifinite if there exists an increasing net of semifinite subfactors $M_i$, $i\in J$, with conditional expectations $E_i:M\to M_i$ ...
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Question about the homogeneity of the state space of a type $\rm{III}_1$ factor
I'm reading the paper Homogeneity
of the State Space of Factors of Type $\rm{III}_1$ by Connes and Størmer.
Homogeneity of the state space means that all normal states are approximately unitarily ...
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Zeta zeros and prolate wave operators
Recently, Connes, Consani and Moscovici in https://arxiv.org/abs/2310.18423 have blended two of their results on zeta zeros and the prolate wave operators, which, they say, "suggests the ...
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Positive cone in Haagerup L²-space: how much information does it contain?
Given a von Neumann algebra $A$, its Haagerup $L^2$-space $H:=L^2A$ (also known as the standard form of the Neumann algebra) comes equipped with a positive cone $P\subset H$.
Question: How much ...
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Can a non-separable C$^*$ algebra have separable GNS Hilbert space
Suppose we have a $C^*$ algebra $\mathfrak{U}$ that is non-separable. Consider a state $ω$ of $\mathfrak{U}$ and the GNS representation $(H_ω,π_ω,Ω)$. Is it possible for $H_ω$ to be separable, and if ...
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On spectral calculus and commutation of operators
Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
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Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
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Strengthening the direct integral decomposition of von Neumann algebas
Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$.
It is ...
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An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible
Given:
$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$
Can you help me come ...
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Bochner theorem for (non-abelian) discrete groups
I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
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Connection between number theory and operator theory
I was wondering if there is any connection between number theory and operator theory.
Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For ...
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Ergodic actions and deviation from invariance
Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
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Does $N \mathbin{\bar{\otimes}} N^{\mathrm{op}}$ act on $L^2(N)$?
Let $N$ be a von Neumann algebra and $N^{\mathrm{op}}$ its opposite. The standard form $L^2(N)$ is an $N$-$N$-bimodule, or equivalently a module over $N \otimes_{\mathrm{alg}} N^{\mathrm{op}}$.
Does ...
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Projections in atomless von Neumann algebras
Let $\varepsilon>0$.
If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
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Order bounded version of monotone complete $C^*$-algebras
Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
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Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism
This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...
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Show that $\mathbb{K}\cong M_{n}(\mathbb{K})$ [closed]
I would like to show the following isomorphy but not sure how to go about this:
$\mathbb{K}\cong M_{n}(\mathbb{K})$
Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism ...
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Comparison between the operator norm and the $L^1$ norm on group algebras
Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question:
The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
- 1,262
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Compare the weight of $p\vee q$ and that of $p+q$
Let $M$ be a von Neumann algebra.
If it has a semifinite faithful normal trace $\tau$, then we have $\tau(p\vee q)\le \tau(p)+\tau(q)$.
However, for the weight (even a faithful normal state) $\omega$ ...
- 617
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Tensor product of faithful normal states is faithful
I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.
I also ...
- 439
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States on Bratteli diagrams
This a reference request. We are writing a paper on calculi on AF algebras and their relation to Dirac operators. This is quite simple for UHF algebras (and we have references), but AF algebras ...
- 1,213
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125
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Transforming nilpotency into diagonalizability [closed]
We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$.
We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows:
$Te_1=0$ and $...
- 3,972
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1
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122
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Unitary representation of a group of automorphism on an abelian algebra
Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
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1
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Lattices and noncommutative algebras in noncommutative geometry
This a question that I've asked in mathematics stack exchange without having received any response :
I am interested in the relation between lattices and noncommutative algebras in the context of ...
- 136
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votes
1
answer
171
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weights of projections and norms of operators in a von Neumann algebra
Let $M$ be an atomless von Neumann algebra equipped with a (semifinite faithful normal) weight $w$. Let $x\in M$ and let $\varepsilon>0$.
Can we find a constant $\delta>0$ such that whenever a ...
- 617
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1
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183
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Hyperfinite factors and increasing fatorization of states
If a factor $R$ contains a matrix algebra $M\subset R$ (i.e., a $M$ is a type $I_n$ factor), then $R \cong M \otimes M^c$ where $M^c=R\cap M'$ is the relative commutant.
Each state $\omega$ on $R$ ...
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