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77 votes
0 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\...
Ady's user avatar
  • 4,060
46 votes
0 answers
2k views

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 $l^2$...
Nik Weaver's user avatar
  • 42.8k
33 votes
0 answers
1k views

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-...
Andreas Thom's user avatar
  • 25.5k
31 votes
0 answers
2k views

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 ...
Mikhail Ostrovskii's user avatar
31 votes
0 answers
1k views

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 ...
Hannes Thiel's user avatar
  • 3,497
27 votes
1 answer
1k views

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 ...
Gary Moon's user avatar
  • 683
27 votes
0 answers
1k views

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 ...
Ali Taghavi's user avatar
26 votes
0 answers
359 views

Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
Joseph O'Rourke's user avatar
24 votes
0 answers
918 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on $...
David Spivak's user avatar
  • 8,659
24 votes
0 answers
751 views

Are amenable groups topologizable?

I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
Łukasz Grabowski's user avatar
24 votes
0 answers
2k views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
user avatar
23 votes
0 answers
1k views

Laplace Transform in the context of Gelfand/Pontryagin

Questions: Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform? If not, is there a ...
Greg Zitelli's user avatar
  • 1,104
22 votes
0 answers
676 views

Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
Vidit Nanda's user avatar
  • 15.5k
21 votes
0 answers
869 views

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 ...
Suvrit's user avatar
  • 28.6k
21 votes
0 answers
732 views

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 ...
Pietro Majer's user avatar
  • 60.5k
21 votes
0 answers
876 views

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 ...
Joel Fine's user avatar
  • 6,247
19 votes
0 answers
552 views

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{...
Samuel Johnston's user avatar
19 votes
0 answers
563 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
Noah Schweber's user avatar
19 votes
0 answers
937 views

What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
Danielle Ulrich's user avatar
19 votes
0 answers
703 views

The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
Boaz Tsaban's user avatar
  • 3,104
18 votes
0 answers
323 views

The analogy between dualizable categories and compact Hausdorff spaces

Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{...
Georg Lehner's user avatar
  • 2,303
18 votes
0 answers
1k views

Does there exist a continuous open map from the closed annulus to the closed disk?

(Originally from MSE, but crossposted here upon suggestion from the comments) In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
D.R.'s user avatar
  • 831
18 votes
0 answers
1k views

"Next steps" after TQFT?

(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.) Recently, I've been ...
Nicholas James's user avatar
18 votes
0 answers
1k views

What is the strongest nerve lemma?

The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology: If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
2xThink's user avatar
  • 81
18 votes
0 answers
370 views

Čech functions and the axiom of choice

A Čech closure function on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$,...
bof's user avatar
  • 13.4k
17 votes
0 answers
677 views

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 ...
Dmitri Pavlov's user avatar
17 votes
0 answers
488 views

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\...
Mikhail Ostrovskii's user avatar
16 votes
0 answers
372 views

On projectively countable sets in the Hilbert cube

A subset $A$ of a topological space $X$ is called projectively countable if for any continuous map $f:X\to\mathbb R$ the image $f(A)$ is countable. It is easy to see that each projectively countable ...
Taras Banakh's user avatar
  • 41.8k
16 votes
0 answers
542 views

$C^*$-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 ...
Yemon Choi's user avatar
  • 25.8k
16 votes
0 answers
1k views

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 ...
Bill Johnson's user avatar
  • 31.5k
15 votes
1 answer
601 views

Topological spaces in which countable intersections of dense open sets have dense interior

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense. Now consider the following strengthening of the Baire ...
Gro-Tsen's user avatar
  • 32.5k
15 votes
0 answers
477 views

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 ...
Dor's user avatar
  • 723
15 votes
0 answers
455 views

Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
15 votes
0 answers
623 views

Is homeomorphism of simplicial complexes semidecidable?

Conventions: $\cong$ is homeomorphism of topological spaces and isomorphism of groups, $\equiv_G$ is the equality of two words over the generators of the group $G$. Simplicial complexes are finite. ...
Ville Salo's user avatar
  • 6,652
15 votes
0 answers
409 views

Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?

Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
Taras Banakh's user avatar
  • 41.8k
15 votes
0 answers
259 views

Spaces locally modelled on $L^2(\mathbb R)$

In this recent question, I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (...
André Henriques's user avatar
15 votes
0 answers
716 views

Is this "Homology" useful to study?

In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can ...
Ali Taghavi's user avatar
15 votes
0 answers
365 views

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 $...
Peter Kosenko's user avatar
15 votes
0 answers
349 views

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 ...
Nik Weaver's user avatar
  • 42.8k
15 votes
0 answers
1k views

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 ...
Marc Nardmann's user avatar
15 votes
0 answers
2k views

Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
14 votes
0 answers
326 views

When can we extend a diffeomorphism from a surface to its neighborhood as identity?

Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
Anubhav Mukherjee's user avatar
14 votes
0 answers
427 views

Which functions have all the common $\forall\exists$-properties of continuous functions?

This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well. For a ...
Noah Schweber's user avatar
14 votes
0 answers
785 views

Covering image of a connected CW-complex need not be a CW-complex

This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
Sumanta's user avatar
  • 632
14 votes
0 answers
343 views

Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
Leo Herr's user avatar
  • 1,084
14 votes
0 answers
860 views

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 "...
Chris Wendl's user avatar
14 votes
0 answers
718 views

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 ...
Tanya Vladi's user avatar
14 votes
0 answers
851 views

Cardinality of the set of continuous functions

Suppose $(X,\tau)$ and $(Y,\sigma)$ are topological spaces. Let $F(X,Y)$ be the set of continuous functions $X\rightarrow Y$. I want to compute the cardinality of $F(X,Y)$. It depends not only on ...
Bugs Bunny's user avatar
  • 12.3k
14 votes
0 answers
543 views

Small cardinals related to topological convergence

Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
Santi Spadaro's user avatar
14 votes
0 answers
205 views

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 ...
Yemon Choi's user avatar
  • 25.8k

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