- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

Section 2.1.9 on Idempotents reads as follows:

https://www.physicsforums.com/attachments/3097

https://www.physicsforums.com/attachments/3098So, on page 43 we read (see above) ...

" ... ... Note that, conversely, a full set of orthogonal idempotents of End(\(\displaystyle M\)) gives rise to a full set of inclusions and projections for \(\displaystyle M\): for each \(\displaystyle i\), take \(\displaystyle L_i = e_iM\), \(\displaystyle \pi_i\) to be \(\displaystyle \pi_i \ : \ \mapsto e_im\) and \(\displaystyle \sigma_i\) to be the evident inclusion map.

I am hoping to fully understand how a full set of orthogonal idempotents of End(\(\displaystyle M\)) gives rise to a full set of inclusions and projections for \(\displaystyle M = L_1 \oplus L_2\) ... but I am unsure what the "evident" inclusion map is? Can someone please help?

Peter