# Questions tagged [banach-algebras]

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### Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
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### Classes of Banach algebras (that aren't operator algebras) whose bidual comes from a “universal representation”

Are there any classes of (Arens regular) Banach algebras that are not operator algebras whose bidual comes from a “universal representation”, as in the case of C*-algebras?
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### Measure algebra on the Bohr compactification vs the bidual algebras

The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it. Let $G$ be a locally compact Abelian group and let $bG$ ...
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### Closed subgroup (Cartan) theorem without transversality nor Lipschitz condition within Banach algebras

Yesterday, I came across the following preliminary theorem. Theorem Let $\mathcal{B}$ be a Banach algebra (with unit $e$) and $G$ be a closed subgroup of $\mathcal{B}^{-1}$ (the group of ...
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### Separability of an algebra is equivalent to separability of its spectrum

Let $A$ be a commutative C*-algebra. I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable. Notes ...
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### 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 $\left|\cdot\right|_X$...
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### Projections in the tensor product of von Neumann algebras

This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here. Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
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### The definiton of a multiplier on a Banach algebra

Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach ...
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\... 0answers 144 views ### Automatic continuity in Banach algebras I have the following two questions 1: Let$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$automatically ... 0answers 155 views ### Gelfand spectrum as a measure space Given a Lebesgue probability measure space$(X,m)$(say, just the unit interval with the Lebesgue measure on it), let$A$be a closed subalgebra of the real$L^\infty(X,m)$. Then one can realize the ... 0answers 51 views ### Evaluate$\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:...
Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...