Articles of banach algebras

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak topology. Please help me. Thanks so much.

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb Z)$?

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. How does it follow that a character is continuous? Is the following a theorem? A ring homomorphism $f: R \to S$ is […]

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ? Actually I try to prove the Spectral representation Theorem of a normal operator in case the representation of $C^*(T)$ over $H$ […]

Two-dimensional unital complex Banach algebras

Let $A_1$ be the matrix algebra consisting of matrices of the form $$ \pmatrix{ \alpha & 0 \\ 0 & \beta\\ }$$ and let $A_2$ be the matrix algebra consisting of matrices of the form $$ \pmatrix{ \alpha & \beta \\ 0 & \alpha\\ }.$$ I want to prove that these algebras are not isomorphic […]

Is $GL(E)$ dense in $L(E)$, when $\dim E=\infty$?

Let $E$ be a normed vector space (Banach space, if you like). Is $GL(E)$, the set of invertible and continuous endomorphism of $E$, dense in $L(E)$, the set of continuous endomorphism of $E$? I specify that I know the answer if $dim(E)<\infty$, with classical arguments about the spectrum of matrices, and, I know that $GL(E)$ […]

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by $\partial_S(\Omega)$ . From the very definiton of peak points we know that every peak point belongs to $\partial_S(\Omega).$ Now the examples that I know the set of peak points coincides with the $\partial_S(\Omega).$ My question here […]

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 chapter 10 of Rudin’s Functional Analysis. Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ […]

The spectrum of a commutative unital Banach algbra having linearly dense idempotents is totally disconnected

I don’t know how should I start to show: If $A$ is an arbitrary abelian Banach algebra in which the idempotents have dense linear span, its specrum (the space of characters on $A$) is totally disconnected.

Example of a singular element which is not a topological divisor of zero

We know that every topological divisor of zero in a commutative Banach algebra is singular. I need an example of a singular element which is not a topological divisor of zero.