Let $A$ be a vn-algebra. Let $e,p$ and $q$ be projections in $A$. Suppose that $p\leq q$. True or false?
$$q\wedge e-p\wedge e\leq q-p$$
2026-03-25 18:59:56.1774465196
A relation between projections
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False in general. Take $A=M_2(\mathbb C) $, $$q=I,\ \ \ p=\begin {bmatrix}1&0\\0&0\end {bmatrix},\ \ \ \ e=\begin {bmatrix}1/2&1/2\\1/2&1/2\end {bmatrix}. $$ Then $e\wedge p=0$, so $$q\wedge e-p\wedge e=e\not\leq q-p. $$