Main question:
Let $X$ be a $\mathbb K$-vector space with $\mathbb K=\mathbb R$ or $\mathbb C$. Let $\mathcal L(X)$ be the set of continous linear applications from $X$ to $X$.
Let $\mathcal A$ be a subalgebra of $\mathcal L(X)$ such that $\{b\in\mathcal L(X)\,|\,\forall a\in\mathcal A,\,ab=ba\}=\{\lambda I\,|\,\lambda\in\mathbb K\}\,.$
I am expecting the following result (but do not know how to prove it).
- Assume $X$ is finite dimensional. Then $\mathcal A=\mathcal L(X)$.
I doubt this is a very original question, could someone indicate me a method to solve this problem?
Some possible extensions:
If it is indeed the case:
- Can we generalize this result to the case of $X$ a Hilbert space, i.e. $\overline {\mathcal A}=\mathcal L(X)$ for some topology to be precised on $\mathcal L(X)$?
- Same thing for $X$ a Banach space.
- And if we replace $\mathcal L(X)$ by a Banach algebra with unity?