# Tagged Questions

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) ...

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### Spectral radius of an element in a Banach algebra

I want to solve this problem: Let $A$ be an unital Banach algebra and $a\in A$ then r(a)<1 \Longrightarrow \lim_{n\to \infty} a^n=0 \Longrightarrow (1-a)^{-1}=\sum_{n=0}^\infty ...
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### The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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### Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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### Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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### Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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### Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
### Rudin functional Analysis chapter $10$, exercise $13$
This is Rudin's functional Analysis chapter $10$, exercise $13$. I am confused about the notation $\sigma_A(f)$, what does that mean?(What role does the subscript $A$ play here). And can someone ...