Let A be a commutative Banach algebra such that set of all characters is infinite. I want to p

aniawnua

aniawnua

Answered question

2022-05-21

Let A be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in A such that its spectrum is infinite.
Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Answer & Explanation

Krish Finley

Krish Finley

Beginner2022-05-22Added 14 answers

Let ( χ n ) n N be a sequence of pairwise distinct characters.
For m n let U m , n = { a A χ n ( a ) χ m ( a ) }. Since U m , n is the complement of the closed hyperplane ker ( χ m χ n ) , the set U m , n is open and dense in A whenever m n. By the Baire category theorem, the countable intersection
D = m n U m , n
is dense and of second category in A, so it is non-empty. Every a D has infinite spectrum because
σ ( a ) { χ n ( a ) n N }
and because the set { χ n ( a ) n N } is infinite by definition of D.
Jaycee Mathis

Jaycee Mathis

Beginner2022-05-23Added 6 answers

I didn't know how to do it either, but now I understand. thanks

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