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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: , , , .

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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.e $$ \phi:C \rightarrow D$$ *-isomorphism .…
lakshit Mehra
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why is every element of the disk algebra A generated by the function $z \mapsto z$

The disk algebra is the set of all functions on the unit disc $D$ which are analytic on the interior of the disc and continuous on the boundary. Addition and multiplication are defined obviously. Why is the disk algebra generated by 1 and the map…
Illustionist
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Prove that $|f|_X\leq \|f\|$ for every $f\in C(X)$

Let $X$ be a compact space and let $\|.\|$ be an algebra norm on $ C(X)$ Show that $|f|_X \leq \|f\|$ for every $f\in C(X)$ Could anyone please suggest me how to deal with these questions
user62498
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Weakly amenable Banach algebras

In Dales' monograph ''Banach Algebras and Automatic Continuity'' there is Proposition 2.8.64 which says that Banach algebra homomorphisms with dense range preserve contractibility as well as amenability. They also preserve weak amenability but under…
Krzysztof
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Characters of $C(X)$

I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$. Now I am wondering if there is a similar identification for $C(X)$ where $X$ is compact Hausdorff with the sup norm?
Student
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Characters in Banach algebras

I am reading Wikipedia and there is something I don't understand: ''Let $A $ be a unital commutative Banach algebra over $\mathbb C$. Since $A $ is then a commutative ring with unit, every non-invertible element of $A$ belongs to some maximal ideal…
Student
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The disk algebra and continuous functions

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the $\sup$-norm. Why this extra restriction to only include $f$…
Student
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Maximal ideals in function spaces

Hello friends of mathematics, i have got a question. In the lecture we proved that the maximal ideals of $C(X)$ are the sets of functions which vanishes on a closed subset of $X$. But now i will look to $C^1[0,1]$ which is a Banach algebra. I will…
Science153
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ultrapowers of a matrix

1)Let A, B, C, D be a Banach algebras, and U be a free ultrafilter. Can we see that ultrapowers of \begin{pmatrix} A & B \\ C & D \end{pmatrix} equal to \begin{pmatrix} (A)_U & (B)_U \\ (C)_U & (D)_U \end{pmatrix}
Albert harold
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Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$

Let $A$ be a Banach algebra with the property $\big(q=pq=qp \Rightarrow \|q\|\leq \|p\|\big)$ whenever $p,q\in A$ are idempotents. Is there a term coined to the algebras with this property in the literature? For an example, $\ell^2$ with pointwise…
Onur Oktay
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A question about Banach algebras: showing that $\operatorname{Sp}a \subset D_o \cup D_1$

Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint. Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let $D_o$ and $D_1$ be the disks in the complex plane of the same…
rese
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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 homomorphism on $l^1(\mathbb Z)$?
Shreya Chauhan
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Invertible products in Banach algebras

I found this interesting challenge: give an example of an unital Banach algebra that contains two elements $x$ and $y$ such that $xy$ is invertible but $yx$ is not invertible. I thought it would be kind of simple to find such an example. Now, after…
ragrigg
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Commutative Banach algebras and maximal ideal space

We know that a unital commutative Banach algebra $\mathcal{A}$ has a compact maximal ideal space $M_{\mathcal{A}}$. But I am curious about the converse. If the maximal ideal space $M_{\mathcal{A}}$ is compact, can we say that the commutative Banach…
yoyo
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Exponential of a matrix, continuous function

Take the Banach algebra $M_n(\mathbb{C})$ and consider the function that for a matrix $A \in M_n(\mathbb{C})$ associates the element $e^A = \sum\limits_{k = 0}^{\infty}\frac{A^k}{k!}.$ Is this function continuous? My attempt: I know that the…
LAU
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