I'm trying to understand the following proof:
I understand the proof entirely except the displayed formula after "Hence we have". Basically I don't see why $$e_{2n}-e_{2n+1} \sim f_{2n+1}-f_{2n+1}.$$ Can someone explain why this must be true?
2026-03-30 16:59:45.1774889985
Doubt in Murray-von Neumann equivalence
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The number 2 is from replacing $n$ with $2n$.
If you put $w=u(e_n-e_{n+1})$, then $$ w^*w=(e_n-e_{n+1})u^*u(e_n-e_{n+1})=e_n-e_{n+1}, $$ $$ ww^*=u(e_n-e_{n+1})u^*=f_{n+1}-f_{n+2}. $$