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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 ...
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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 ...
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### 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 ...
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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. ...
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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 ...
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1 answer
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### 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. ...
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### 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 $C^\ast$ ...
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1 answer
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### 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 ...
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### 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 ...
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### 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 ...
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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$[\cup_{n\geq 0} \chi_{-n} H^\infty$]...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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