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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).

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Higher dimensional Goursat theorem and Cauchy theorem possible?

This might not be a proper question. But I will try my best to present my ideas. If $A$ is a real unital Banach algebra with finite dimension ($>1$), but not necessarily a division algebra. If a ...
user760's user avatar
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$A \otimes J+ I \otimes B$ is prime ideal of $A \otimes B$.

Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Let $I$ and $J$ be prime ideals of $A$ and $B$ respectively. The following fact should be easy but I ...
Math Lover's user avatar
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0 votes
1 answer
46 views

Non-commutative banach algebra

A theorem states that if $A$ is a commutative Banach algebra, then for all $a\in A$ we have $$\sigma(a)=\{\chi(a) : \chi \text{ is a character }\}$$ My question what if $A$ is not commutative. Does ...
vemapo's user avatar
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1 vote
0 answers
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Shilov boundary of unit sphere

Let $B=\{z=(z_1, z_2) : |z_1|^2+|z_2|^2<1\}$ denote the unit ball in $\mathbb{C}^2$ and let $\partial B$ be its topological boundary, i.e. the unit sphere in $\mathbb{C}^2$. One of the texts by J. ...
Curious's user avatar
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Contraction and isometry

Let $X$ and $Y$ be two untial commutative Banach algebras. Suppose $X\subset Y$ and $X$ countractively embeds $Y$. Why is it not the case that the embedding is isometric? From what I see, since $X$ is ...
Toasted_Brain's user avatar
0 votes
1 answer
28 views

Every closed ideal of a Banach *-algebra with approximate identity is semiprime?

An ideal $I$ of an algebra $\mathcal{A}$ is called semiprime if $J^2 \subseteq I$ implies $J\subset I$ for all ideals $J$. It is known that every closed ideal of a $C^{\ast}$-algebra is semiprime. ...
Math Lover's user avatar
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2 votes
1 answer
75 views

Applying the Stone-Weierstrass Theorem to an example

I know the Stone-Weierstrass Theorem: If $X$ is compact and $\mathcal{A}$ is a closed subalgebra of $C(X)$ such that: $1 \in \mathcal{A}$ for $x,y \in X$ with $x \neq y$, there exists f$ \in \mathcal{...
Philip's user avatar
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1 vote
1 answer
29 views

Multiplicative linear functional on noncommutative unital Banach algebra is bounded and norm-decreasing

It is a standard theorem that, for an abelian unital Banach algebra, every nonzero multiplicative linear functional is bounded and has norm at most $1$. I don't see why the proof can't work word-by-...
user760's user avatar
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0 votes
1 answer
58 views

Norm Inequality in Banach Algebra

Suppose $\mathcal{A}$ is a unital Banach Algebra and P,Q are two closed ideals of $\mathcal{A}$ such that $P\subseteq Q$. Then can we say $||x+P||\leq ||x+Q||\quad \forall x\in \mathcal{A}$ ? I tried ...
Damini's user avatar
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0 answers
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Homotopy between bounded linear operators on Banach spaces

In my course of Functional Analysis, talking about continuity method and index of Frendholm operators we construct homotopy between operators. However i encountered some difficulties to connecting ...
Manuel Bonanno's user avatar
3 votes
1 answer
59 views

Unital Banach algebra as a quotient of a free complex algebra

Let $\mathbb{C}\langle x_i:i\in I\rangle$ be the free complex noncommutative unital algebra over the generators $X:=\{ x_i:i\in I\}$. Then, every unital complex algebra $A$ is a quotient of $\mathbb{C}...
user760's user avatar
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1 answer
82 views

Commutator Ideal of Toeplitz Operators

Suppose $H^{2}$ is a Hardy space and $T_{\phi}$ is Toeplitz operator on $H^{2}$ with symbol $\phi\in L^{\infty}(\mathbb{T})$. $\mathcal{A}$ is a C*-algebra generated by $\{T_{\phi}:\phi\in L^{\infty}(\...
Halmos's user avatar
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2 votes
1 answer
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Examples of The Gelfand Representation Not Behaving

Currently I am trying to think of examples of Banach algebras $A$ where the Gelfand Map $\Gamma: A \longrightarrow C_0(\Omega(A))$ fails to be an isomorphism. So far, I have come up with some mundane ...
Isochron's user avatar
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3 votes
1 answer
66 views

Idempotent of a Banach algebra on the boundary of an ideal must belong to the ideal?

Let $A$ be a Banach algebra and $J$ be an ideal of it. If $1 \in \partial J,$ then there exists some point $x\in J$ such that $\|1-x\|<1/2,$ which means $x$ is invertible, and hence $1\in J.$ I am ...
Ma Joad's user avatar
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1 answer
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Zero-divisors in a unital $C^*$-algebra

I have been studying the functional calculus for unital $C^*$-algebras. This says the following. Let $A$ be an unital $C^*$-algebra and $a \in A$ a normal element. We denote by $B:= C^*(1, a) \subset ...
Anton Odina's user avatar
2 votes
1 answer
69 views

Prove in a C* algebra that a*a is positive

I am trying to find a reference to the following "obvious facts" (not sure if they are true or not, but should have some comparable similar results) regarding a non-commutative $C^\ast$ ...
Ma Joad's user avatar
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0 votes
1 answer
62 views

Is the spectrum of an element in an algebra, $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$? [closed]

In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that $\sigma(x) = \{z \in\mathbb{C}: \vert z\vert\leq\Vert x \Vert\}$ But the proof is given only for one side ...
Easwaran N's user avatar
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41 views

Eigenvalue Problem for the Laplace-Beltrami Operator in Real Hilbert Spaces

Usually, Spectral Theory for Operators in Hilbert Spaces is defined over $\mathbb{C}$, the complex field. I'm trying to study some results of Spectral Theory in the real field. Specifically, I have ...
ayphyros's user avatar
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1 answer
63 views

Is any ideal of a Banach algebra closed?

This is the proof I have obtained to show that any ideal of a Banach algebra is closed: If $\cal B$ is a Banach algebra, then let $I$ be an ideal (bilateral). Let $\{y_n\}_{n\in{\mathbb N}}$ a ...
stkcpc's user avatar
  • 21
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0 answers
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About the algebra $H^\infty + C(\mathbb{T})$

We know that $H^\infty + C(\mathbb{T})$ is the closed subalgebra of $L^\infty(\mathbb{T})$ containing $H^\infty$. How to show that $H^\infty + C(\mathbb{T})$ = clos$[\cup_{n\geq 0} \chi_{-n} H^\infty$]...
Halmos's user avatar
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1 vote
0 answers
31 views

example of a $B$-bimodule $X$ that is not an $A$-bimodule.

Suppose that $(A, ∥ · ∥_A)$ and $(B, ∥ · ∥_B)$ are two Banach algebras such that $B$ is an ideal in $A$ and $∥·∥_A ≤ ∥·∥_B$. We know that each Banach $A$-bimodule $X$ is also a Banach $B$-bimodule. ...
A. Friend's user avatar
1 vote
1 answer
39 views

Continuity of adjoint of algebra homomorphism

Consider unital (complex) abelian Banach algebras $A$ and $B$ with corresponding ideal spaces $\Sigma_A$ and $\Sigma_B$. Suppose $\rho:A\to B$ is a continuous algebra homomorphism that maps $1_A$ to $...
Oskar Vavtar's user avatar
1 vote
1 answer
35 views

Uniqueness of non-zero homomorphism in abelian Banach algebra

Consider $\mathbb{C}^3$ equipped with product $\bullet$ given as $$(x_1,x_2,x_3)\bullet(y_1,y_2,y_3)~=~ (x_1y_1,x_1y_2+x_2y_1,x_1y_3+x_2y_2+x_3y_1)$$ and norm $\|(x_1,x_2,x_3)\|=|x_1|+|x_2|+|x_3|$. $A:...
Oskar Vavtar's user avatar
1 vote
0 answers
118 views

Define multiplication on $L^1(\mathbb{R})$

I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
InnocentFive's user avatar
1 vote
0 answers
72 views

For a self-adjoint element $a$ in a unital Banach $*$-algebra, is it true that $\|a\| \mathbb{1} -a \geq 0$?

In a Banach $*$-algebra positive elements can be defined as finite sums of the sort $\sum\limits_k b_k^*b_k$, is it true then that, for a self-adjoint $a$, the expression $\|a\|\mathbb{1} -a$ is a ...
Felipe Dilho's user avatar
0 votes
1 answer
47 views

Character space of abelian Banach algebra is weak* closed in the unit ball of dual space.

Say $A$ is an abelian Banach algebra and $\Omega(A)\subset A^* := B(A, \mathbf{C})$ is the set of non-zero algebra homomorphisms. Consider now $\Omega(A)$ with the weak$^*$ topology. I am trying to ...
Anton Odina's user avatar
3 votes
1 answer
109 views

$\mathcal{C}^2[0,1]$ is a Banach Algebra

The following is problem 13 here. Consider functions in $\mathcal{C}^2[0,1]$ and $a,b>0$. In this case, if we define: $$\lVert f \rVert:=\lVert f \rVert_\infty+ a \lVert f' \rVert_\infty +b \lVert ...
Kadmos's user avatar
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0 votes
2 answers
62 views

Existence of a non-zero continuous function vanishing at infinity.

I am currently reading the book '$C^*$-Algebras and Operator Theory' by Gerard J. Murphy and I have trouble understanding two statements on page 4. Let $\Omega$ be a locally compact Hausdorff space ...
Anton Odina's user avatar
0 votes
1 answer
37 views

Property of the Gelfand transform

I'm reading this script and I got stuck on the last Let $X$ be the set of all non-zero multiplicative linear functionals on the unital commutative Banach algebra $\mathcal{A}$ then $\mathrm{sp}(\hat{A}...
Davide Modesto's user avatar
2 votes
1 answer
127 views

Exponential boundedness of a strongly continuous semigroup $(T_t)_{t>0}$.

Let $T = (T_t)_{t>0} $ be a strongly continuous semigroup (of bounded operators) on a Banach space $E$, i.e, $ \lim_{t \rightarrow z} \|T_tx- T_{z}x \|, \forall z >0, \forall x \in E $. Note ...
Jeffrey Jao's user avatar
0 votes
0 answers
48 views

An isolated point in the spectrum must be a point in the approximate point spectrum??

Is the following claim and proof correct?? I can't seem to find an error. Let $X$ be a complex Banach space, and $A\in \mathcal{B}(X)$ a bounded operator. Then, if $\lambda\in \sigma(A)$ is isolated ...
user760's user avatar
  • 1,670
1 vote
1 answer
64 views

Definition of Poles in spectral theory

In usual complex analysis, a pole $z_0\in \mathbb{C}$ of a function $f$ that is holomorphic on a punctured disk $0<|z-z_o|<R$ is defined as a zero of function $\frac{1}{f}$. But in spectral ...
user760's user avatar
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0 votes
0 answers
45 views

$L^1(G)$ is a Banach *-Algebra [duplicate]

Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
Pedro Lourenço's user avatar
0 votes
1 answer
55 views

Trying to understand the functional calculus of a unitary operator.

I am trying to understand the theorem on this Wikipedia article. It is stated as follows. Theorem. Let $x$ be a normal element of a $C^*$-algebra A with an identity element e. Let $C$ be the $C^*$-...
caffeinemachine's user avatar
0 votes
1 answer
41 views

Need help understanding proof that $\hat{x} (\hat{A}) = \sigma(x)$

From Functional Analysis: Spectral Theory, by V. S. Sunder, page 97. Let $A$ denote a unital commutative Banach algebra, then $\hat{x} (\hat{A}) = \sigma(x)$, where $\hat{A}$ is the collection of all ...
user avatar
0 votes
1 answer
66 views

Inequality between exponential in unital Banach Algebra

So, I am trying to solve the following statement: Suppose that $\sigma(a) \subseteq \{\lambda \in \mathbb{C}: \text{Re} \lambda < 0\}$. Show that there exist $M, \omega \in \mathbb{R}$ with $\omega ...
Tipeg's user avatar
  • 118
0 votes
0 answers
31 views

Is the given space is a banach algebra and find it’s maximal ideal space?

Let $ \mathcal{A} =\left\{f \in C(\mathbb{T}): \forall n \in \mathbb{N}^{+}, \int_{0}^{2 \pi} f\left(e^{i \theta}\right) e^{i n \theta} \mathrm{d} \theta=0\right\}$.prove $ \mathcal{A}$ is a banach ...
hicozhao's user avatar
6 votes
2 answers
478 views

Example of operator that commutes with a given multiplication that is not itself a multiplication.

I am aware that given a diagonal matrix $D$ whose diagonal entries are all distinct, any matrix $A$ that commutes with $D$ must be itself diagonal. I am also aware that this result does not generally ...
SecretlyAnEconomist's user avatar
3 votes
1 answer
219 views

Decomposition of the spectrum of a closed Banach subalgebra

In the book An Introduction to Operator Algebras by Kehe Zhu the following theorem is stated: Theorem. Suppose $A$ is a closed subalgebra of a unital Banach algebra $B$ with $\hat{1}\in A$. If $x \in ...
ferolimen's user avatar
  • 630
2 votes
1 answer
115 views

Maximal ideal space of an $L^\infty$ space is extremely disconnected

I'm reading the book An Introduction to Operator Algebras by Kehe Zhu, in a such book all the algebras are assumed to be unital and associative. Once said that, the autor proves that if $A$ is a ...
ferolimen's user avatar
  • 630
1 vote
1 answer
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Let $I$ be a semiprime ideal of $\mathbf{A}$. Suppose $a\mathbf{A}a \subseteq I$ for some $a$. Show that $a \in I$.

Let $\mathbf{A}$ be an algebra. An ideal $I$ of $\mathbf{A}$ is called semiprime if $\mathbf{A}/I$ has no nonzero nilpotent ideal. Can someone supply a proof of following fact? Let $I$ be a ...
Math Lover's user avatar
  • 3,662
3 votes
1 answer
205 views

Maximal ideal space of the $n$-dimensional ball algebra

Let $B_n \subset \mathbb{C}^n$ be the $n$-dimensional open ball of radius $1$ centered at the origin. Let $A$ be the set consisting of holomorphic functions in $B_n$ which are continuous in $\overline{...
SCarlsen's user avatar
  • 135
0 votes
0 answers
29 views

How to salvage this quick proof of the exponential sum law in Banach algebras

From the power series definition of the exponential function in the real or complex numbers you can quickly prove the sum rule by considering the function $$ f(x) = \exp(a + b - x) \exp(x), $$ proving ...
sudgy's user avatar
  • 131
1 vote
1 answer
37 views

Resolvent inequality follows it's boundness

Following is from my lecture notes. I have resolvent map $R(\cdot,x): \mathbb{C} \rightarrow A$, where $x\in A$ Banach algebra. Domain is whole complex numbers because for the sake of contradiction it ...
user3342072's user avatar
1 vote
0 answers
64 views

Banach subalgebra generated by commutative set equals the centralizer of its centralizer

Let $A$ be a unital Banach algebra and $S\subset A$ a subset consisting of commuting elements. Let $C(S)$ be the centralizer of $S$, that is, $y \in C(S)$ iff $ys=sy$ for all $s \in S$. I have already ...
ferolimen's user avatar
  • 630
4 votes
0 answers
68 views

The set of homomorphism on $C_\infty(X)$ for a locally compact $X$

Let $X$ be a locally compact Hausdorff space, which is not compact. $C_b(X)$ is a Banach algebra of all bounded continuous functions with the sup norm. Let $C_\infty(X)$ be the Banach algebra of ...
Anna  Vakarova's user avatar
1 vote
1 answer
62 views

The set of unital homomorphisms

Let $X$ be a locally compact Hausdorff space, which is not compact. $C_b(X)$ is a Banach algebra of all bounded continuous functions with the sup norm. Let $C_\infty(X)$ be the Banach algebra of ...
Anna  Vakarova's user avatar
2 votes
0 answers
42 views

Prove that dual of a unital Banach algebra is nonempty using resolvent and Liouville's theorem

Let $\mathcal{A}$ be a unital Banach algebra over complex numbers $\mathbb{C}$. For every $a \in \mathcal{A}$, let $\sigma(a)$ be the spectrum of $a$. Define the resolvent of $a$ to be $ R(a,z) = (...
Anna  Vakarova's user avatar
0 votes
1 answer
110 views

The algebra of all bounded linear operators acting on a complex Banach space is a prime algebra.

Let $X$ be a Banach space over the complex field $\mathbb{C}$ and $\mathcal{B}(X)$ the algebra of all bounded linear operators on $X.$ I want to show that $\mathcal{B}(X)$ is a prime algebra. My ...
Akhter's user avatar
  • 3
2 votes
1 answer
76 views

Connected component of the constant function $1$ in $C(S^1,\mathbb{C})$

Let $C(S^1,\mathbb{C})$ be the space of continuous functions $f:S^{1} \subset \mathbb{C} \longrightarrow \mathbb{C}$ endowed with the supremum norm, that is, $ \lVert f \rVert= sup_{z \in S^{1}} |f(z)|...
ferolimen's user avatar
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