2
votes
1answer
90 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
0
votes
0answers
32 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
3
votes
1answer
42 views

Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
1
vote
1answer
28 views

simply polar elements in a ring

An element $a$ in a ring $A$ with identity is said to be simply polar if there is $b$ for which $a=aba$, with $ab=ba$. If in addition $b=bab$ then such an element $b$ is unique. The question is ...
1
vote
1answer
80 views

Prime ideal for the Banach algebra

The maximal ideal and Jacobson radical often appear in the Banach algebra theory, but I do not see the prime and nilradical in it. We can define a prime for a Banach algebra following the ring ...
4
votes
2answers
95 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 ...