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# Questions tagged [banach-algebras]

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### positive functional on Banach *-algebra (with appro. identity) is continuous?

Theorem (N. Th.Varopoulos): Let $\mathcal{B}$ be a Banach *-algebra with a bounded approximate identity. Then every positive functional $T$ on $\mathcal{B}$ is continuous. I think this theorem is ...
1 vote
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### An example of a non rigid Banach algebra

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain ...
194 views

### Closed prime ideal in $C[0, 1]$

I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal. Is there any $\textbf{closed}$ prime ...
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### Bound for the product of Sobolev functions in $W^{s,1}$

I would like to bound the product of two functions $f$, $g$ in the space $W^{s,1}$. $$\lVert fg\rVert_{W^{s,1}}\leq c\lVert f \rVert \lVert g \rVert.$$ It seems reasonable to want to use Hölder's ...
1 vote
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### About isometric Banach algebra isomorphisms and WAP functionals

Let $B$ be a Banach algebra and $A\subseteq B$ a subalgebra. It is known that there is a quotient map $\phi: B^*/\textrm{wap}(B)\to A^*/\textrm{wap}(A)$ that is also an $A$-bimodule map. Let's say ...
1 vote
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### Representations of the dual Banach algebra pair $(\ell_1,c_0)$

Let $\displaystyle E_p=(\bigoplus_{n\in\mathbb{N}} \ell^1_n)_{\ell^p}$ for some $1<p<\infty$ and $\ell^1 = \ell^1(\mathbb{N})$ be equipped with the convolution. Then, there exists an isometric &...
1 vote
102 views

### 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 ...
1 vote
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### About vector valued measure algebras

Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra. Is $M(G,A)$ a Banach algebra (with convolution as the ...
1 vote
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### Algebras sitting inside reproducing kernel Hilbert space other than multiplier algebra

Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space. If the kernel is normalized then the multiplier algebra $\mathcal{M}$ is an algebra that is sitting inside $\mathcal{H}$. Is there any ...
56 views

### Weierstrass subdomain of $\DeclareMathOperator\Spm{Spm}\Spm \mathbb{Q}_p$

I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$. Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with ...
326 views

### Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009. Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
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### If $A\hat\otimes B$ has identity then so are $A$ and $B$

Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the ...
### $B(X)$ and $B(X)^{**}$ for superreflexive Banach spaces $X$
Let $A$ be a Banach algebra. It is well-known that $A^{**}$ with either Arens product has separately weak$^{*}$ continuous multiplication if and only if $A$ is Arens regular [Theorem 1, ...
A Banach algebra $A$ is a dual Banach algebra if it is a dual Banach space with a (not necessarily unique) predual $A_{\ast}$, and the multiplication on $A$ is separately weak*-continuous. Dual Banach ...