All Questions
3,601 questions with no upvoted or accepted answers
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Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
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208
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Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain
I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
- 2,670
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318
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Is there any connection between deformation theory in algebraic geometry and perturbation theory in functional analysis/PDEs?
Particularly, is there any connection between formal/first-order/infinitesimal deformation theory and perturbation theory? Both subjects involve "perturbing" some structure at a point, so ...
- 1,094
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182
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Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
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241
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Tomita–Takesaki theory and subfactors
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $\Omega$ be a cyclic and separating vector in $H$. Let $J$ and $\Delta$ be the corresponding modular conjugation and modular ...
6
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122
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Premeasurability of affiliated operators for type $\textrm{III}$ von Neumann algebras
$\DeclareMathOperator\dom{dom}$If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action ...
- 7,067
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Condensed/liquid vector spaces and path integrals
[Edited to take into account comments.]
Background
One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
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208
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Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?
Let $f_i: [0, 1] \to \mathbb R$ be functions in $L^1 \cap L^\infty$ with $\sup_i \|f_i\|_{L^\infty} < M$ for some $M > 0$.
Suppose $f_i$ converge weakly in $L^1$ to some $L^1$ function $f$ - ...
- 6,215
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290
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Two questions about Fock spaces
Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
- 3,515
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124
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Meagre sets of bounded operators
Let $H$ be a separable, infinite-dimensional Hilbert space and let $\mathbb{B}(H)$ be the algebra of bounded operators on $H$. The norm topolology on $\mathbb{B}(H)$ is stricly finer, hence the ...
- 9,853
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113
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A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
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132
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Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
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107
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Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
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What are some results that assume the Connes' embedding conjecture or any of its reformulations?
As you all (may) know, the Connes embedding conjecture was disproven last year. Also, as its Wikipedia page shows, there are multiple reformulations (but it is definitely not an exhaustive list):
...
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533
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Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
- 341
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153
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How absolute is NIP in a model?
The following question is motivated by a model theoretic question but doesn't really involve any model theory per se. That said, I don't know the appropriate keywords for the relevant functional ...
- 12.5k
6
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241
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Extension of positive functionals
Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
- 6,035
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107
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Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
- 3,816
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158
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Quotients of subspaces of $C(\alpha)$
A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
- 986
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348
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Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user153636
6
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144
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Does every locally convex space with a Schauder basis have the approximation property?
For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well ...
- 799
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99
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Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
- 11.3k
6
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315
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Question on operator topologies convergence
Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ ...
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321
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Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
- 596
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181
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Blocksum induces a unital H-space structure on the space of Fredholm operators
Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
- 386
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239
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Sheaves on Rectifiable Sets
Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...
6
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113
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Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
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6
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117
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Homomorphisms from BV
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...
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233
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Group $C^*$ vs group von-Neumann algebras
Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...
- 9,853
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237
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A characterisation of certain $C^*$-algebras
I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
- 633
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Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle
I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature.
I'm hoping there's a nice ...
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83
views
Are invertible measures strictly dense?
Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
- 113
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169
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Characterizing fullness of a von Neumann algebra by the topology of its bimodules
Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map:
$$
\mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
- 1,090
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132
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Schröder–Bernstein for representations of operator algebras
This is claimed in a Wikipedia Article:
If two representations (of a $C^*$-algebra $A$) $\rho$ and $\sigma$, on Hilbert spaces $H$ and $G$ respectively, are each unitarily equivalent to a ...
- 18.7k
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88
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Density of squares of radial eigenfunctions
The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
- 8,086
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282
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Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
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4k
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Interchange of supremum and integral
Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.
Are there some results under which the following interchange ...
- 303
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240
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Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
- 493
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388
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Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
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137
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Spectrum of perturbed differential operators
I am looking for a reference that could help me with the following two questions:
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
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281
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Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
- 519
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798
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What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user14166
6
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210
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Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
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774
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
- 167
6
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272
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Extension operators for topological vector space-valued smooth functions on closed sets
There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
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6
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252
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Constructing Extreme Points in Reflexive Banach Spaces
A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
- 1,073
6
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63
views
Chord-arc property of n-tuples of commuting operators
Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a ...
- 315
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369
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Interpolation between $H^1$ and $H^1\cap L^1$
Suppose that $T:H^1(\mathbb{R}^3)\rightarrow\mathbb{R}$ is a linear bounded operator, with operator norm $M_2$. In particular, given $1\leq p\leq2$, there exist optimal constants $M_p\leq M_2$ such ...
- 943
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216
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Integral-like concepts
I am looking for interesting concepts (I guess you could say functionals from the function space $[a;b]\to\mathbb R$) that are like integrals in some respect.
The background is that I have proven a ...
- 1,241
6
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118
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Categorical description of dense homomorphisms of topological algebras
Let $A$ and $B$ be topological associative algebras (no matter, in which sense, for example, over $\mathbb C$, with identity, and with separately continuous multiplication). Let us say that a (...
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