Let $A$ a complex Banach algebra containing a unit $e$. For $x \in A$, you can define the spectrum $\sigma(x)$ and the resolvent set of $x$.
However, if you consider a complex Banach space $X$ and a bounded linear operator $T:X\to X$, then the spectrum of $T$ can be divided into the parts: point spectrum of T, approximate point spectrum of $T$ and residual spectrum of $T$.
What can you do if you want such a decomposition in case of $A$? Elements of $A$ have to act on a fixed Hilbert space $H$, but there are problems in which everything depends on $H$. Is there any other possibility to obtain such a decomposition of elements of a complex Banach algebra A?
I hope my question becomes clear.
Regards