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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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Cc(X) is Banach Algebra

Let X be a compact Hausdorff space. Then Cc(X) is a commutative Banach Algebra with infinity norm. I tried by showing Cc(X) a sub algebra of C(X). I maneged to show that Cc(X) is a vector subspace of C(X) but how to show Cc(X) is closed under ring…
user578405
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In Banach algebra $A$ find an example such hat $e^{a+b} \not =e^ae^b$

Let $A$ be a Banach algebra if $ab=ba$ then we have $e^{a+b}=e^ae^b$ without $ab=ba$, I want to find an example such hat $e^{a+b} \not =e^ae^b$ Any help will be greatly appreciated.
user62498
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In Banach algebra $A$ if $ab=ba$ prove that $e^{a+b}=e^ae^b$

Let $A$ be a Banach algebra if $ab=ba$ then prove that $e^{a+b}=e^ae^b$ I've started by $e^a=\sum _{n=0}^{\infty }\frac{a^n}{n!}$, I want to know if this is correct way? Any help will be greatly appreciated.
user62498
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$C^*$ algebra, existence of particular state

If we have $a\in A$ be arbitrary element of $C^*$ algebra $A$. Can we find a faithful state $\phi$ such that $\phi(a) = k$ for $k$ in $spec(a)$?
Sushil
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Existence of Bounded Approximate Identities for Modules of a Normed Algebra

In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given: Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ \{ e(\lambda) \} $ in $ A $ such that $…
LMW
  • 445
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Gelfand Spectrum on non-unital banach algebra

Let A be a non-unital Banach Algebra. $\Omega(A)=\{$set of all non zero multiplicative linear functionals of $A$ $\}$ I want to prove that $\Omega(A)\cup\{0\}$ is weak* compact. I've been given a hint that we have to show that the one point…
user141561
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Misleading wikipedia entry on Banach algebras

On the wikipedia page for banach algebras, under examples it states: The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions. and under counterexamples it states The algebra of the…
Mike Izbicki
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how to show following is totally disconnected?

We know that $l_{\infty}$ be banach algebra ( under pointwise multiplication and with sup norm). How to show that its maximal ideals space is totally disconnected?
user195218
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convergence of approximate unit

Let A be a Banach algebra and I be a closed ideal.let $\{e_\alpha\}_\alpha$ be an approximate identity for I.Prove that for all $ a \in A $,$\{ae_\alpha\}_\alpha$ is convergence (or example the violation).
mehrdad
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$*$-isomorphism on $L(E)$

Suppose $M_2$ be the set of all $2\times 2$ matrices and $L(M_2)$ be the set of all adjointable operators from $M_2$ to $M_2$. Can you mention an example of$~~$$*-$isomorphism operator from $L(M_2)$ to $L(M_2)$? (not be identity map)
Alireza
  • 153
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If $X$ is compact Hausdorff, is every isomorphicm $\mathcal{C}(X) \to \mathcal{C}(X)$ continuous?

If $X$ is a compact Hausdorff space, is then every isomorphism from ${\mathcal C}(X)$ onto ${\mathcal C}(X)$ is continuous?
user221746
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Is this true? If ‎$‎\varphi\in ‎\mathcal{B}_{‎\mathcal{A}}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}_{‎\mathcal{A}}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection, why?. Such that $E$ and $F$ are right Hilbert ‎$‎‎‎\mathcal{A}‎$‎-modules and $\mathcal{B}_{‎\mathcal{A}}(E,F)‎$‎ is the set…
Alireza
  • 153
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Does this define a norm s.t. set of matrices is Banach algebra?

As I asked in this comment here: If $a$ is an $n \times n$ matrix and we define $$ \|a\|_p = \left ( \max_j \sum_i |a_{ij}|^p \right)^{1 \over p}$$ or $$ \|a\|_p = \left ( \max_i \sum_j |a_{ij}|^p \right)^{1 \over p}$$ Does this norm make the set…
user167889
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Showing that $\|1\| = 1.$

Let $A$ be a Banach algebra. A map $a \mapsto a^*$ is called an involution if for all $a,b \in A$ and $\lambda \in \mathbb C,$ $(1)$ $(a^*)^* = a,$ $(2)$ $(a + b)^* = a^* + b^*,$ $(3)$ $(\lambda a)^* = \overline {\lambda}\ a^*,$ $(4)$ $(ab)^* = b^*…
RKC
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Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules?

Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules? clearly $E=F$ are $C^*$-algebra
Alireza
  • 153
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