Questions tagged [banach-algebras]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
79 votes
4 answers
7k views

Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I: The theory of commutative ...
Jonas Meyer's user avatar
  • 7,279
78 votes
3 answers
8k views

Norms of commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
Bill Johnson's user avatar
31 votes
1 answer
2k views

Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?

Here's a research problem, which I think interesting. Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...
Narutaka OZAWA's user avatar
22 votes
5 answers
1k views

Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
BigbearZzz's user avatar
  • 1,245
21 votes
2 answers
2k views

In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
Malik Younsi's user avatar
  • 1,942
18 votes
3 answers
1k views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ \...
Leonard's user avatar
  • 816
17 votes
1 answer
1k views

Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
Mohammad Golshani's user avatar
15 votes
1 answer
858 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ \mbox{...
Eusebio Gardella's user avatar
15 votes
1 answer
925 views

Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
Nate Ackerman's user avatar
15 votes
1 answer
671 views

Open bilinear maps that are not uniformly open

A map $f\colon X\to Y$ between metric spaces is uniformly open whenever for each $\varepsilon >0$ there is $\delta >0$ such that for any $x\in X$ one has $$B_Y\big(f(x),\delta\big)\subseteq f\...
Tomasz Kania's user avatar
  • 11.3k
15 votes
0 answers
354 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
341 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
14 votes
4 answers
517 views

About the existence of characters on $B(X)$

Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$? I know the proof of the fact that $M_n(\mathbb{C})$ ...
User93709's user avatar
  • 355
14 votes
2 answers
918 views

C*-algebras with bizzarre structure of projections

This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C*-algebra such that its projections do not form a lattice (...
Jan Veselý's user avatar
14 votes
3 answers
3k views

The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$

Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
yeshengkui's user avatar
  • 1,373
14 votes
1 answer
651 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
Jochen Glueck's user avatar
14 votes
2 answers
546 views

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $\ell^1(G)$ is a flat (right) $\mathbb{Z}G$-module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by-finite group. Denote the ...
Jiang's user avatar
  • 1,518
12 votes
2 answers
576 views

Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...
M.González's user avatar
  • 4,301
12 votes
0 answers
448 views

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, ...
Tomasz Kania's user avatar
  • 11.3k
11 votes
2 answers
2k views

Operator that commutes with projections

We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$ Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
Sascha's user avatar
  • 506
11 votes
2 answers
544 views

Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by $(zf)(g):...
Werner Thumann's user avatar
11 votes
1 answer
1k views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
Anindya Biswas's user avatar
10 votes
3 answers
4k views

Projections in Banach spaces

Dear All, I am absolutely lost in the following problem: Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
zoltan.leka's user avatar
10 votes
1 answer
1k views

Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
Fedor Goncharov's user avatar
10 votes
1 answer
374 views

Maximal ideals of the ring $\mathbb C \{T\}$

Consider the Banach $\mathbb C$-algebra $$ \mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace $$ With the norm given by $\| \sum a_i T^i\| = \sum |...
Aitor Iribar Lopez's user avatar
10 votes
0 answers
783 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
Torsten Schoeneberg's user avatar
10 votes
0 answers
196 views

Projective tensor squares of uniform algebras

In discussion with a colleague recently (Jan 2017), $\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$ I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
Yemon Choi's user avatar
  • 25.5k
10 votes
0 answers
505 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
TrzyTrypy's user avatar
  • 101
9 votes
2 answers
295 views

Two inequalities in $C^*$ algebras

Under what conditions on a $C^*$ algebra $A$ we have the following inequality: $$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$ The second identity which I am looking for is ...
Ali Taghavi's user avatar
9 votes
2 answers
968 views

Which Banach algebras are group algebras?

Given a locally compact Hausdorff group $G$, one can construct several Banach star-algebras using $G$ (and its associated Haar measure): $L^1 (G)$, $M(G)$ (regular complex measures on $G$), $L^{\infty}...
Mark's user avatar
  • 4,804
9 votes
3 answers
334 views

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^...
David Gao's user avatar
  • 1,262
9 votes
2 answers
306 views

Explicit proof that $c_0$-module $\ell_\infty$ is not projective

It is well known in narrow circles that the homological dimension (in the sense of relative Banach homology) of $c_0$-module $\ell_\infty$ is 2. As the corollary, this module is not projective. This ...
Norbert's user avatar
  • 1,687
8 votes
3 answers
1k views

is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here: I know that in general the sum and product of normal elements need not be normal. It is even true ...
Tom's user avatar
  • 967
8 votes
3 answers
1k views

Conceptually, what does unitization do?

Let $(\mathcal A,||\cdot||)$ be a normed algebra (with or without a unit). The unitization of $\mathcal A$ is the space $\mathcal A_+:=\mathcal A\oplus \Bbb C$ where the multiplication operation $\...
BigbearZzz's user avatar
  • 1,245
8 votes
3 answers
2k views

Measures on general topological groups

I am interested in the group algebras of non-locally compact groups. What references can you advise? This is a wide question, so I list more concretely what I would like to see: Here X can be even ...
Yulia Kuznetsova's user avatar
8 votes
2 answers
333 views

Non-density of invertible elements in $\ell_1(\mathbb{N}_0)$

Consider the Banach algebra $\ell_1(\mathbb{N}_0)$ (with convolution / Cauchy product of series). I am looking for an elementary proof of the fact that the group of invertible elements in this algebra ...
Tomasz Kania's user avatar
  • 11.3k
8 votes
1 answer
334 views

characterization of commutative Banach algebras

Let $A$ be a Banach algebra with the following property: For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. ...
Ali Taghavi's user avatar
8 votes
1 answer
298 views

What algebras are quotients of $\ell_1(\mathbf{N})$?

Every separable Banach space is a linear quotient of $\ell_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell_1(G)$ for some group $G$ (these are the so called ...
user512365's user avatar
8 votes
1 answer
294 views

Masas in second duals of Banach algebras

Lel $B$ be a Banach algebra and give $B^{**}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding ...
Cleft's user avatar
  • 203
8 votes
1 answer
616 views

When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?

Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form $$ u(A)=\...
Sergei Akbarov's user avatar
8 votes
1 answer
423 views

Injectivity for bimodules and Hochschild cohomology

Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
user avatar
8 votes
1 answer
1k views

Spectral mapping theorem for polynomials in $z,\overline{z}$ and direct construction of the function calculus for a normal operator

Suppose that $A$ is an element in Banach algebra and $p$ is a polynomial. Then we have an equality $p(\sigma(A))=\sigma(p(A))$ where $p(A)$ has an elementary meaning. This theorem (the spectral ...
truebaran's user avatar
  • 9,140
7 votes
3 answers
399 views

Are nearby subalgebras of matrix algebras conjugate?

Let $k=\mathbb{R}$ or $\mathbb{C}$ and let $A$ be a finite-dimensional $k$-algebra. If $A$ is simple, then the Skolem-Noether theorem says that any two algebra homomorphisms $f, g: A \to M_n(k)$ are ...
Matthias Ludewig's user avatar
7 votes
2 answers
894 views

English translation of Hilbert's work

Does anyone know if there is an English translation of Hilbert's: "Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig, 1912". ?? Thanks, Andre
Andre's user avatar
  • 111
7 votes
3 answers
799 views

Preduals of B(E)

For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators. If $E$ is a Banach space then is it known whether $B(E)$ is always a ...
Ollie's user avatar
  • 1,391
7 votes
1 answer
666 views

$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$

Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum $A=\sum \mathcal{K}(\mathcal{H})$ of countably many ...
Habujew's user avatar
  • 113
7 votes
1 answer
522 views

Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be (...
Arundhathi's user avatar
7 votes
1 answer
376 views

Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra?

Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra? (Strong morita equivalence is the same ...
Louis A's user avatar
  • 350
7 votes
1 answer
685 views

When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: ...
Norbert's user avatar
  • 1,687
7 votes
1 answer
713 views

Banach Algebra Counterexample

Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ? (if possible, can you add a proof or a reference ? ) Thank you very much ! ...
jpp's user avatar
  • 73

1
2 3 4 5
8