3
votes
1answer
47 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
4
votes
1answer
66 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
0
votes
1answer
73 views

Continuity of a map from $\mathbb{C}$ to a Banach algebra

Consider the map from $\mathbb{C}$ to a unital Banach algebra $B$ given by $x \mapsto \exp(xb)$ for $b\in B$. I proved that this map is continuous by using the definition of $\exp(xb)$ as a contour ...
3
votes
2answers
203 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
2
votes
1answer
299 views

spectral radius.

I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem? The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ...