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Questions tagged [banach-algebras]

371 questions
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1 answer

Whether every algebra norm $\left|\cdot\right|$ on $C(X)$ is equivalent to uniform norm $\left|\cdot\right|_X$

Suppose that $X$ be a compact space and $\left|\cdot\right|$ be an algebra norm on $C(X)$ Is every algebra norm $\left|\cdot\right|$ on $C(X)$ equivalent to uniform norm $\left|\cdot\right|_X$? I don't know where to start. Any clues?
user62498
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2
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Banach algebras for which left invertible implies invertible

Are there noncommutative Banach algebras in which left invertibility implies invertibility? If so, what are they called?
Andre
  • 111
2
votes
1 answer

Two completely different norms on a unital algebra!

Does there exist any unital normed algebra $(A,\|\cdot\|)$ enjoying another norm $\|\cdot\|_1$ such that $(A,\|\cdot\|_1)$ forms a unital normed algebra with the same unit. Any element contained in the intersection $$ \{x\in A :…
ABB
  • 3,898
2
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Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric. Is there a separable Banach algebra that contains isomorphic images of all separable Banach algebras? Is there a commutative…
Robert Hermann
  • 63
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Is $L_{\infty}(G)^{**}$ a without order Banach algebra?

A Banach algebra A is without order if for all $x \in A$, $xA=\{0\}$ implies $x=0$, or, for all $x \in A$, $Ax=\{0\}$ implies $x=0$. If $G$ is a compact abelian group then $L_{\infty}(G)$ is a Banach algebra with the convolution product and its…
El nota
  • 21
1
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A dual Banach algebra question

Let $\Gamma$ be an infinite discrete abelian group and $A=\ell^1(\Gamma)$ denote its group algebra. Clearly, $A_*=c_0(\Gamma)$ is a predual of $\ell^1(\Gamma)$ for which $(A,A_*)$ is a dual Banach algebra. By Matthew Daws' Representation Theorem,…
Onur Oktay
  • 2,263
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hulls of non-closed prime ideals

If P is a closed prime ideal in a commutative unital complex Banach algebra, then the hull of P is connected (an easy consequence of Shilov's idempotent theorem) Is the same true for non-closed prime ideals?
ray
  • 687
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Automatic continuity in Banach algebras

I have the following two questions 1: Let $A$ and $B$ be Banach algebras and suppose that $B$ is semisimple. Let $T:A \to B $ be a homomorphism with $\overline {TA}=B.$ Is $T$ automatically continuous? 2: Let $A$ and $B$ be Banach algebras. Let…
user62498
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Evaluate $\operatorname{Rad}(A/\operatorname{Rad}(A))$ in a Banach algebra

I've asked this question here Let $A$ be a Banach algebra with identity $e_A$, I'd like to find $\operatorname{Rad}(A/\operatorname{Rad}(A)).$ whre we define $\operatorname{Rad}(A)=\{a\in A:e_A-ba \in \text{InvA},b\in A\}$ I think it's equal…
user62498
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Spectrum space of semidirect product of a subalgebra and an ideal of a Banach Algebra

If an algebra $A$ is a semidirect product of a subalgebra $B$ and an ideal $I$. Is characterized the character space of $A$ by character space of $B$ and character space of $I$?
Ali
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Quotient of a Banach algebra

Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?
Albert harold
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