The positive part of an element in a von Neumann algebra

Question

When reading the proof of lemma, I met with some troubles.$A_2=\sum_{n=1}^{\infty}(t_n+1)E_n$, how to check that $\tau((A_2–I)_{+})=\sum_{n=1}^{\infty}t_n$?

$(A_2–I)_{+}=(A_2-I)\chi_{[0,\infty)}(A_2-I)$, how to prove that $(A_2–I)_{+}=\sum_{n=1}^{\infty}t_nE_n$?

Answer 1

We have $$ A_2-1=\sum_{n=1}^{\infty} (t_n+1)E_n - (\sum_{n=1}^{\infty} E_n + (\vee_{n=1}^{\infty} E_n)^{\perp})=\sum_{n=1}^{\infty} t_n E_n - (\vee_{n=1}^{\infty} E_n)^{\perp}. $$

For any $m$, projections $E_m$ and $(\vee_{n=1}^{\infty} E_n)^{\perp}$ are orthogonal, and it follows that

$$ (A_2-1)_+=\sum_{n=1}^{\infty} t_n E_n. $$