# 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).

928 questions
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### The set of all inner derivations $\mathcal{B}^1(A, X)$ is not closed in the set of continuous derivations $\mathcal{Z}^1(A, X)$,

Let $A$ be a complex Banach algebra and $X$ be a Banach $A$-module The set of all continuous $X$-derivations on $A$ is a complex linear space, denoted by $\mathcal{Z}^1(A, X)$. and The set of all ...
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### When does the spectrum of an element in a Banach algebra with involution lie in the open right half-plane?

Let $A$ be a Banach algebra with involution, $x\in A$ and $t\in {\mathbb R}$ such that $t>\rho(xx^*)$. Show that $\sigma(te-xx^*)$ lies in the open right half-plane. I have no idea! It's obvious ...
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### Gelfand map is topologically injective

Let $A$ be a commutative Banach algebra and let $$G: A \rightarrow C_{0}(\text{Spec}(A))$$ be a topologically injective map. Recall that the map $T: X \rightarrow Y$ is called topologically ...
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### In Banach algebra $A$ if $a_n\to a$ then $e^{a_n}\to e^a$

My question is as follows let $(a_n)$ be a sequence in banach algebra $A$ and $a_n\to a$ then prove or disprove $e^{a_n}\to e^a$ Any help will be greatly appreciated
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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 ...
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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 $spec(a)$?
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### If ${A}/{I}$ and $I$ , where $I$ is a closed ideal, have approximate identities then so does $A$.

Let $A$ be a Banach algebra and $I$ a closed ideal in $A$. I want to prove that If ${A}/{I}$ and $I$ have left approximate identities then $A$ has a left approximate identity Any help will be ...
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### Approximate identity in $\ell _p$

Show that for $1\leq p<\infty$, $\ell _p$ with multiplication defined by $(a_n)_n(b_n)_n=(a_nb_n)_n$has an unbounded approximate identity but no bounded approximate identity, I don't know how to ...
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### Can C*-envelope introduce the unity?

Let $A$ be a non-unital Banach $*$-algebra with isometric involution. Is it possible that the enveloping $C^*$-algebra of $A$ is unital? I guess not at least when $A$ admits an approximate identity, ...
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### In a unital Banach algebra $A$ with $ab=ba$ prove that $\sigma(a+b)\subset \sigma(a)+\sigma(b)$ [duplicate]

Let $A$ be a unital Banach algebraIn with $ab=ba$ prove that $\sigma(a+b)\subset \sigma(a)+\sigma(b)$ and $\sigma(ab)\subset \sigma(a)\sigma(b)$
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### Showing uniqueness of holomorphic functions on Banach spaces using uniquess of scalar holomorphic functions?

Theorem Let $D ⊆ C$ be a connected open set, and let $z_n ∈ D$, $n ∈ \mathbb{N}$, be a sequence which converges to a point $z_0 ∈ D$ such that $zn \neq z_0$ for all $n ∈ \mathbb{N}$. Further let $E$ ...
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### In Banach algebra $A$ if $a^2=a$ show that $\sigma_A(a)\subseteq\{0,1\}$

Let $A$ be a Banach algebra with identity $e$ and let $x\in A$ be such that $a^2=a$ show that $\sigma_A(a)\subseteq\{0,1\}$ and compute the resolvet function $$R(a, \lambda)=(\lambda e-a)^{-1}$$ ...
Suppose that $A$ and $B$ are two Banach algebra with identity and $T:A \to B$ be an isomorphism prove that $\sigma _{A}(a)=\sigma _{B}(T(a))$ for all $a\in A$ My atempp: Let \$\lambda \not\...