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Questions tagged [banach-algebras]

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that $|xy| ≤ |x||y|$. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use or instead (or in addition). Further related tags: , , , .

1383 questions
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Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?

As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in complex analysis: every bounded, entire function…
Mark
  • 5,952
57
votes
3 answers

The identity cannot be a commutator in a Banach algebra?

The Wikipedia article on Banach algebras claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is surprising to me, but maybe the proof is…
Qiaochu Yuan
  • 468,795
38
votes
1 answer

How to justify solving $f(x+1) + f(x) = g(x)$ using this spectral-like method?

Let's say that I want to find solutions $f\in C(\Bbb R)$ to the equation $$ f(x+1) + f(x) = g(x) $$ for some $g\in C(\Bbb R)$. I can write $f(x+1) = (Tf)(x)$ where $T$ is the right shift operator and rewrite the equation suggestively as $$ (I+…
BigbearZzz
  • 15,530
32
votes
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Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to famous theorems. E.g. his collected works contain…
Torsten Schoeneberg
  • 29,325
23
votes
3 answers

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative linear map? Or, is there some obstruction,…
Rasmus
  • 18,946
21
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1 answer

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = \mathbb{C}$. Liouville's theorem for Banach spaces says…
Qiaochu Yuan
  • 468,795
20
votes
1 answer

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor find a counter example. I wonder whether anyone…
Hui Yu
  • 15,469
19
votes
2 answers

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ is an open set, and if $E$ is finite dimensional…
Davide Giraudo
  • 181,608
17
votes
1 answer

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D\colon A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense in $A$, and any derivation $D\colon A\to I$…
Hamid Shafie Asl
  • 3,025
16
votes
0 answers

Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ are unconditionally…
mr_e_man
  • 5,986
15
votes
1 answer

On the spectrum of the sum of two commuting elements in a Banach algebra

Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que $a*b=b*a$. Pourquoi $σ (a+b) \subset σ(a)+σ(b)$. Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)? Translation: Let $A$ be a unitary Banach algebra and the…
masmoudihoussem
  • 348
14
votes
1 answer

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
Philipp G. Sinicyn
  • 901
14
votes
1 answer

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to participate, and who have never taken a course in functional…
Daniel Moskovich
  • 4,254
14
votes
1 answer

Proof of Gelfand formula for spectral radius

STATEMENT: Let $A$ be a Banach algebra, then for every $x\in A$ we have $$\lim_{n\rightarrow\infty}\|x^n\|^{1/n}=r(x)$$ Proof: We know that $r(x)\leq \lim \inf_n\|x^n\|^{1/n}$, so it suffices to prove that $$\limsup_{n\rightarrow…
Enigma
  • 3,989
13
votes
1 answer

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have $\phi(ab)=\phi(a)\phi(b)$. The set of all character is shown by…
Hamid Shafie Asl
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