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).

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2
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0answers
41 views

If $\ell^1(S)$ is unital, must the semigroup $S$ have an identity?

Assume $S$ is non-empty. If $\ell^1(S)$ is unital, then there exists $e\in\ell^1(S)$ with $f\star e=e\star f=f$ for all $f\in\ell^1(S)$ and $\|e\|_1=1$. Let $a\in S$. Then $\delta_a\in\ell^1(S)$ so ...
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9 views

Associativity of convolution in $\ell^1(S)$, where $S$ is a semigroup.

This is what I have done but I'm not sure if it is sufficiently rigorous. Let $f,g,h\in\ell^1(S),t\in S$. Then $$((f\star g)\star h)(t)=\sum_{rs=t}(\sum_{uv=r}f(u)g(v))h(s)=\sum_{rs=t}\sum_{uv=r}f(u)g(...
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Prove that Banach algebra C(X) isomorphic to the disk algebra $\mathcal{A}$ for some compact Hausdorff space X if and only if $\text{int}X=\emptyset$

Prove that Banach algebra C(X) isomorphic to the disk algebra $\mathcal{A}$ of continuous functions on the disk $D = \{z \mid |z| \leq 1\}$ which are holomorphic on the interior of $D$, with the sup ...
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compact subset of $\mathbb{C}$ is spectrum. (Banach space / hilbert space)

I'm studying that what set can be the spectrum of operator. If $\mathcal{H}$ is an infinite dimension Hilbert space and $K$ is a non-empty compact subset of $\mathbb{C}$, show that there is an $A$ in ...
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1answer
36 views

Proving that $\|f*g\|_1\leq\|f\|_1\|g\|_1$ for all $f,g\in\ell^1(\mathbb{Z})$?

Let $f,g\in\ell^1(\mathbb{Z})$, where $$\ell^1(\mathbb{Z}):=\lbrace f:\mathbb{Z}\rightarrow\mathbb{C}:\sum_{n=-\infty}^\infty|f(n)|<\infty\rbrace$$ and convolution is defined as: $$(f*g)(n):=\sum_{...
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1answer
30 views

Closure of Laurent polynomials is convergent Laurent series?

Let $\mathbb{T}:=\lbrace z\in\mathbb{C}:|z|=1\rbrace$ and $A$ be the subalgebra of $C(\mathbb{T})=C(\mathbb{T};\mathbb{C})$ consisting of the Laurent polynomials $\lbrace\sum_{k=-n}^na_kz^k:a_k\in\...
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25 views

banach algebra and modular left ideal

I'm solving conway's functional analysis. Let $\mathcal{A}$ be a Banach algebra but do not assume it has an identity. If $I$ is a left ideal of $\mathcal{A}$, we say $I$ is modular left ideal if ...
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9 views

Algebra of polynomials generated by $ a_1, a_2, \cdots, a_n \in B (H).$

Let $ H $ be a Hilbert space and $ a_1, a_2, \cdots, a_n \in B (H) $ bounded operators. Let's define, $$A:={\{\sum_{finite}\alpha_{k_1,\cdots,k_n}^{N_1,\cdots,N_n}\alpha_{k_1}^{N_1}\cdots\alpha_{k_n}^{...
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1answer
24 views

Banach algebra 's homomorphic functional

If $\mathscr{X}$ is a Banach algebra over $\mathbb{C}$, is it every homomorphism $\phi : \mathscr{X} \rightarrow \mathbb{C}$ continuous? $\mathscr{X}$ can have a identity and commutative if needed. ...
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23 views

Ways of defining multiplication with corresponding exponentials and logarithms for vector spaces

I am looking at different ways of defining multiplication for vectors, and how to construct an exponential and logarithmic function given that definition. I'm sure this must be an explored subject, ...
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1answer
23 views

Proof that $C_0(X)$ is closed under addition and multiplication

In the lecture notes I have, $C_0(X)$ is defined as follows: Let $X$ be a locally compact, Hausdorff topological space. Then $C_0(X)$ is the set of all continuous complex-valued functions on $X$ that ...
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37 views

Unitary operator as cayley transform of self adjoint

Using functional calculus of bounded functions is easy to see that the application of the function $t \to \frac{t-i}{t+i}$ to any self-adjoint operator $T$ gives a unitary operator $U$ whose spectrum ...
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1answer
17 views

Is an invertible element in a Banach algebra always within a certain neighborhood of another invertible element?

There is a well-known result in functional analysis saying that if $S$ is an invertible bounded operator on a Banach space $V$ and $T\in \mathcal L(V)$ verifies $\left\|S-T\right\|<\dfrac{1}{\left\|...
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1answer
41 views

Multiplicative Limits in Banach Algebra

Let $(a_n), (b_n)$ be sequences in a Banach algebra and $(c_n) := (a_n.b_n)$ If $\lim (c_n) = c$ and $\lim (a_n) = a $ does it follow that $(b_n)$ has a limit $b$ satisfying $c = a.b$ ? I can see this ...
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16 views

Topological zero divisors of $\mathbb C^n$

We know that every zero divisor is a topological zero divisor but not every topological zero divisor is a zero divisor. I was trying to find topological zero divisors in $\mathbb C^n$. Suppose $x=(x_1,...
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1answer
58 views

Maximal ideals of $C^1[0,1]$

What are the maximal ideals of $C^1[0,1]$? We know that the maximal ideals of $C[0,1]$ are of the form $\{f:f(x)=0\}$ and we use the compactness of $[0,1]$ to prove this,but how do we find maximal ...
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32 views

Commutative monoid structures on $\Bbb{N}$

Suppose $m \oplus n$ is a commutative and associative binary relation $\oplus: \Bbb{N} \times \Bbb{N} \to \Bbb{N}$, and that $1$ is an identity element for this operation. In other words, $(\oplus, \...
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0answers
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Banach algebra structure of $\ell^p$ sequence space

Long intro: questions after examples. The space of absolutely convergent sequences $\ell^1 = \{ \vec{a} = (a_1, a_2, a_3, ...) \in \Bbb{R}^\Bbb{N}: \sum_{n} |a_n| < \infty \}$ is closed under ...
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Finding radical of $l^1(\mathbb Z)$

Find the radical of $l^1(\mathbb Z)$ Now we know that radical of a Banach algebra is the intersection of kernels of all $\phi$ where $\phi$ is a nonzero complex homomorphism. But how do I find complex ...
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23 views

Matrices of C* algebra

Let $A$ be a $C^*$- algebra and consider space of matrices on $C^*$-algebra for fix natural no. $n$ $M_n(A) = \{[a_{ij}]_{i,j=1}^{n}: a_{ij}\in A \ \ \ \ for \ \ \ 1 \leq i, j \leq n\}$. We have ...
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*-isomorphism on C* algebra

In case of unital $C^*$ algebras $A$ and $B$ we have that any $^*$-isomorphism preserves norm. Now my question is that what if we take non-unital $C^*$ algebras does $^*$-isomorphism preserves norm i....
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1answer
17 views

Checking whether $\mathbb C^n$ is semisimple or not

Consider $\mathbb C^n$ as a Banach algebra over $\mathbb C$. Is it semisimple? A Banach algebra is semisimple if the intersection of all maximal ideals of it is ${0}$. I can't figure out what are the ...
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41 views

For any two positive operator $a,b$ with norm $<1$ there is a $c$ such that $a,b\leq c$ and $\|c\|<1$. Does this hold for $\|a\|=\|b\|=1$?

I mean, for two positive operator $a,b$ with norm $1$, is there still always a positive $c$ such that $a,b\leq c$ and $\|c\|=1$? Suppose it is in a non-unital C*algebra. And $a\leq b$ means $b-a$ is ...
2
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1answer
71 views

Is $\mathcal{L}(V)^*$ path connected?

Assume that $V$ is a complex Banach space, $\mathcal{L}(V)$ is the Banach algebra composed of bounded linear operators with the norm $\lVert A\rVert=\sup\{\lVert Av\rVert:\lVert v\rVert\leq 1 \}$. $$ ...
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1answer
27 views

Quotient algebra is involutive: do we need the ideal do be self-adjoint?

Let $A$ be a $*$-algebra, i.e. an algebra $A$ together with a map $*: A \to A$ such that $$(a+ \lambda b)^* = a^* + \overline{\lambda} b^*$$ $$a^{**}= a$$ $$(ab)^* = b^* a^*$$ My book then claims that ...
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4answers
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Why is $f(t) = e^{ta}$ differentiable in a unital Banach algebra?

Let $A$ be a unital Banach algebra. For $a\in A$, we define $$\exp(a):= \sum_{n=0}^\infty \frac{a^n}{n!}$$ Consider the function $$f: \Bbb{R} \to A: t \mapsto \exp(ta) = \sum_{n=0}^\infty \frac{t^n a^...
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1answer
21 views

Spectrum of an element in the disk algebra.

Consider the disk algebra $A$ of continuous function on the unit disk $D$ that are analytic on the interior of the disk. Is it true that $\sigma_A(f) = f(D)$ for $f \in A$? A simple yes or no suffices....
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2answers
46 views

Restricting a function in the disk algebra

Let $A$ be the disk algebra, i.e. continuous functions on the closed unit disk in $\Bbb{C}$ that are analytic on the interior of the disk. By the maximum-modulus theorem, we have an isometric ...
1
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1answer
22 views

Spectrum of an upper triangular matrix.

Consider the algebra $A$ of upper triangular matrices. Given $$a= \begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n}\\0 & a_{22} & \dots & a_{2n}\\ 0 & 0 & \dots & a_{...
2
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1answer
43 views

Two definitions of $C_0(X)$. Do they coincide?

Let $X$ be a topological space. Then we can define $$C_0(X):=\{f \in C(X)\mid \forall \epsilon > 0: \exists K \subseteq X \mathrm{\ compact}: \forall x \notin K: |f(x)| < \epsilon\}$$ If $X$ is ...
2
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1answer
31 views

A simple question about boundary of spectrum

Let $A$ be a Banach algebra with identity and $G$ is set of all invertible elements of $A$. $\sigma(x)=\{z\in \Bbb C : ze-x\ \textrm{is not invertible} \}$ is spectrum of $x\in A$ where $e$ is ...
0
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1answer
45 views

Cauchy Product of more than two series

From Wikipedia (https://en.wikipedia.org/wiki/Cauchy_product#Generalizations): $$\prod_{j=1}^n \left( \sum_{k_j = 0}^\infty a_{j,k_j} \right)=\sum_{k_1 = 0}^\infty \sum_{k_2 = 0}^{k_1} \cdots \sum_{...
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0answers
47 views

Generator of Banach algebra $C(\partial D )$

The following problem is a Exercise 18, section 8, chapter 7 of Conway's A Course in Functional Analysis. I want to show that Banach algebra $C(\partial D )$ ($\partial D$ the boundary of $D=\{z:|z|\...
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1answer
29 views

Invertibility of an element in a Banach algebra (Gelfand's formula)

In Folland's A Course in Abstract Harmonic Analysis, Theorem 1.8 states that for a unital Banach algebra (with unit $e$), the spectral radius of an element $x$ is given by $\lim_{n \to \infty} \|x^n\|^...
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1answer
57 views

Is there a natural square root for a path of operators?

Suppose you have a smooth path of invertible operators $t \rightarrow A(t) $ such that $A(0) = Id$. Is there a way of defining a natural square root $X(t)$ such that the following identity holds ? $$\...
2
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2answers
96 views

Prove that $C[0, 1]$ is NOT Approximately Finite

The following question is from $C^*$- Algebras by Example written by Kenneth R. Davidson. The original question is Problem III.6 in exercises after Chapter 3. $\mathit{Definition}$: A $C^*$- ...
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2answers
29 views

Banach Algebra: $r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2$

I am trying to see that if we have a complex Banach algebra with unity, we will have that $$r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2.$$ I was able to do the first implication: Since we know that $\|x^2|...
2
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1answer
27 views

Show that each $\hat x$ is a member of $C_0(\Delta)$

Suppose $A$ is a commutative Banach algebra without a unit, let $\Delta$ be the set of all complex homomorphisms of $A$ which are not identically $0$. Each $x\in A$ defines a function $\hat x$ on $\...
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2answers
43 views

Volterra Integral Operator

Let $T: C([a,b])\rightarrow C([a,b])$ be the Volterra Integral Operator, where $T(\phi)(t)=\int_a^tk(t,s)\phi(s)ds$. I have already seen that this operator is compact using the Ascoli-Arzela Theorem. ...
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0answers
3 views

find a non continuous complex homomorphism in non complete normed algebras

I want to show by an example there is a non continuous complex homomorphism in non complete normed algebras Can anyone give me an example? Thank you!!
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0answers
30 views

What exactly is the relationship between weak* convergence and sequential convergence?

I am studying Banach algebras, and the theory frequently makes use of the weak* topology which I am not very familiar with. As I understand it, the weak* topology is defined as follows: let $X$ a ...
2
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1answer
34 views

Prove the uniqueness of the hereditary $C^*$ algebra generated by a positive element

The following question is from $C^*$-Algebras by Example written by Kenneth R. Davidson. The original question is the Problem I.11. $\mathit{Definition}:$ Say $\mathcal{W}$ is a $C^*$-subalgebra of ...
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0answers
30 views

Spectrum of a Banach algebra VS spectrum of a $C^*$-algebra

The spectrum of a $C^*$-algebra $A$ is the set of unitary equivalence classes of irreducible $*$-representations of $A$. The spectrum of a unital commuative Banach algebra $B$ is the set of ...
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1answer
14 views

Redundant assumption in proving the quotient of a Banach algebra over a modular ideal is a field

When reading Murphy's book $C^*$ algebras and operator theory, I found the Lemma 1.3.2 a bit strange. The original lemmas is as the following: If $I$ is a modular maximal ideal of a unital abelian ...
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0answers
25 views

Work verification and alternative solution: sufficient condition for a Banach algebra to be commutative

I have to prove the following: Let $A$ be a Banach algebra with unit $e.$ If there exists $M>0$ such that $\Vert xy \Vert \leq M \Vert yx \Vert$ for all $x, y \in A,$ then the function $\lambda \...
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1answer
28 views

Introduction to Banach Spaces and Algebras, by Allan, Exercise 4.6

I am reading Introduction to Banach Spaces and Algebras, by Allan, Unfortunately I do not succeed in completing, Exercise 4.6 Let $(A, \|\cdot\|)$ be a normed algebra. A Banach algebra $(B, \||\cdot|\...
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0answers
34 views

Are bounded operators on a Banach space a $C^*$-algebra?

It is well-know that if $H$ is a complex Hilbert space then the set of all bounded operators $\mathcal{B}(H)$ is a $C^*$-algebra. If we change $H$ with a complex Banach space $E$, do we have the same ...
2
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0answers
50 views

$C^n[a,b]$ is a Banach algebra

Prove that $C^n[a,b]$ with pointwise multiplication of functions and a norm defined as $\displaystyle\|x\| = \sum_0^n \dfrac{\|x^{(k)}\|_{\infty}}{k!}$ is a Banach algebra. As I understand I need to ...
2
votes
1answer
41 views

Spectrum of unital, commutative C star algebra

According to the Wikipedia article on the Gelfand Represenetation (C* algebra section), the spectrum of a commutative C* algebra $A$ (the non-zero *homomorphisms $\phi : A \rightarrow \mathbb{C}$) i)...
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0answers
33 views

Meaning of “analytic function” in the context of Banach algebras

In going through Folland's Abstract Harmonic Analysis, I came on the following. Let $\mathcal{A}$ be a unital Banach algebra with unit $e$, and define $$\sigma(x) = \{ \lambda \in \mathbb{C} : \lambda ...

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