Let $A$ be a von Neumann algebra, and $T$ be an hermitian element of $A$. Show that the spectral projections of $ T $ belong to $A$.
Proof: the spectral projections of $ T $ commute with every operator that commutes with $T$. so the spectral projections of $ T $ belong to $A$.
My question: Why do the spectral projections of $ T $ commute with every operator that commutes with $T$ and conversely ?