2
votes
1answer
23 views

Local Module Homomorphism

Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have ...
3
votes
1answer
57 views

Module homomorphism

Let $A$ be a Banach algebra with norm $\|.\|_A$ and $X$ be a Banach space with norm $\|.\|_X$. If there exists a operation $.:A\times X\to X$ such that for any $a,b\in A$ and $x,y\in X$ we have ...
2
votes
1answer
47 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
1
vote
0answers
71 views

Commutative Noetherian Banach algebra.

Prove that: 1) Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. 2) Every commutative real unital Noetherian Banach algebra ...
1
vote
1answer
60 views

Banach algebra: norm distance of non-invertible elements to unit element

Let $\mathcal A$ be a commutative, unital Banach algebra. Take $A \in \mathcal A$ such that $A$ is non-scalar, i.e. $A\neq \alpha \mathbb I $, where $\mathbb I$ is the unit element. Denote the ...
6
votes
1answer
237 views

On the spectrum of the sum of two commuting elements in a Banach algebra

Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a. Pourquoi σ (a+b) с σ(a)+σ(b) Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)? Translation: Let ...
4
votes
2answers
94 views

Commutativity in a Unital Banach Algebra

Let $ A $ be a unital Banach algebra and $ S $ a non-empty subset of $ A $. The centralizer of $ S $ is defined as $$ Z(S) \stackrel{\text{def}}{=} \{ a \in A ~|~ \text{$ as = sa $ for all $ s \in S ...
2
votes
1answer
143 views

Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...