# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

3,277
questions with no upvoted or accepted answers

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### 2, 3, and 4 (a possible fixed point result ?)

The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...

46
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0
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### Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...

33
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### Subalgebras of von Neumann algebras

In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...

31
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### When are two C*-algebras isomorphic as Banach spaces?

We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...

30
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### Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...

27
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1
answer

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### The dual of $\mathrm{BV}$

$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...

27
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### Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...

21
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857
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### Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...

21
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719
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### Closed connected additive subgroups of the Hilbert space

It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...

21
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864
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### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

19
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### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...

17
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### Talagrand's "Creating convexity" conjecture

We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...

17
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1
answer

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### Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...

17
votes

0
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479
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### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...

17
votes

0
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### Finite Rank Commutators

My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...

16
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536
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### $C^*$-algebra generated by those operators that are bounded on every $\ell_p$

Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...

15
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469
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### Quantitative Skorokhod embedding

The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...

15
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0
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355
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### Admissible relations in a Banach algebra

Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...

15
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### Is there support for the term "Gelfand algebra"?

In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law
($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be ...

15
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### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

14
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356
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### Are dualizable topological vector spaces finite-dimensional?

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...

14
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714
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### strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...

14
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### Lower bounds on analytic functions connected to Fox H

The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...

14
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0
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203
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### Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?

A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...

14
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0
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627
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### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

14
votes

0
answers

3k
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### Tanh version of a Fourier Transform?

I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...

14
votes

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answers

2k
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### Schwartz kernel theorem for A-linear operators

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...

13
votes

0
answers

690
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### Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...

13
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0
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379
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### Converse to Riesz-Thorin Theorem

Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....

13
votes

0
answers

365
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### Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...

13
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301
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### Banach spaces with $d(X,Y) = 1$

We recall that the Banach-Mazur distance between two isomorphic Banach spaces is given by $d(X,Y) = \inf \{ \|T\| \|T^{-1}\| : T$ is an isomorphism from $X$ to $Y\}$.
It is a classical result that we ...

13
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0
answers

322
views

### Kolmogorov width for cartesian products

For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is
$$
\delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...

13
votes

0
answers

464
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### Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?

(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard bold.)$\newcommand{\FA}{{...

13
votes

0
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558
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### Symmetric (extended) Haagerup tensor product

Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...

13
votes

0
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814
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### How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...

12
votes

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539
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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 ...

12
votes

0
answers

131
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### Existence of more than two C*-norms on algebraic tensor product of C*-algebras

Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...

12
votes

0
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194
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### UMD constant of finite dimensional spaces

For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...

12
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### Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...

12
votes

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338
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### Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...

12
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0
answers

452
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### Is there a simple and reflexive Banach algebra?

There are many Banach algebras which, as Banach spaces, are reflexive. Of course, unitisation is just adding one dimension so this operation preserves reflexivity, hence there are many reflexive, ...

12
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452
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### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

12
votes

0
answers

849
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### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

12
votes

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272
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### Star-shaped Folner sequence

Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that $...

12
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0
answers

468
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### Where to use differential calculus on space of measures?

One great inside of Felix Otto is that the Wasserstein metric from optimal transportation gives the space of (finite second moment, probability) measures on $\mathbb{R}^n$ (or a manifold) a kind of ...

12
votes

0
answers

431
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### Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...

12
votes

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470
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### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

11
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334
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### Tauberian Theorem for 1-parameter groups of operators

The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...

11
votes

0
answers

645
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### What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?

The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions:
$\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...

11
votes

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253
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### Quantifier swap in Banach space theory

The uniform boundedness principle and its corollaries from a logical point of view are statements of when one can swap quantifiers in Banach spaces. Take for instance the principle of condensation of ...