I am reading 'K-Theory' by Max Karoubi and am having difficulty understanding some statements in the section I.6.22. (p.33-34) The set-up is as follows : Let $A$ be an arbitrary banach algebra, and let $M$ be a finitely generated projective module over $A$. Thus there is a surjection $A^m \to M$. Karoubi then talks about the norm induced on $M$ by this surjection.
I am not sure whether I understand it correctly. My understanding is that since $M$ is projective, there is a section of the above surjection. Thus after making a choice of section, $M$ can be thought of as a subspace of $A^m$, and inherits a norm from the latter.
However I am not sure whether this line of reasoning is correct, since in general there would be many choices of sections and the norm might depend upon the particular choice of section. While the language in the book (use of the word 'the' instead of 'a') seems to imply that there is a canonically induced norm.
Also it is not clear whether different norms induced by different choice of sections would lead to different topologies or the same topology on $M$.
Am I missing something ?
Noting $\pi$ the surjection, the norm $\|\cdot\|_{\pi}$ in question on $M$ is defined as $\|\pi(v)\|_{\pi} = \inf_{v'\in \pi^{-1} (\pi (v))} \|v'\|$ where $\|\cdot\|$ is a chosen norm on $A^m$. Note that $\|\cdot\|_{\pi}$ is a norm if and only if $\pi$'s kernel is closed in $A^m$.
When you stress the "the" in "the norm induced on M by this surjection", note that this norm is not canonical at all, as it depends on $\pi$ and on the norm chosen on $A^m$. Even for same type of norm (max or sum let's say for instance), it is not obvious at all that the quotient norms (provided $\pi$'s are continuous) will all be the same.