Projections: Orthogonality

Question

Given a unital C*-algebra $1\in\mathcal{A}$.

Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$

Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$

Then equivalently: $$P\perp P'\iff 0=PP'=P'P\iff\Sigma P^2=\Sigma P=\Sigma P^*$$

How can I check this?

(Operator algebraic proof?)

Answer 1

I'm assuming that by $\sigma(\Sigma P)\leq1$ you mean that $\|\Sigma P\|\leq1$.

1. Suppose that $\|P+Q\|\leq1$. So $0\leq P+Q\leq 1$. Then $(P+Q)^2\leq P+Q$ (just conjugate with $(P+Q)^{1/2}$). That is, $$ P+Q+QP+PQ\leq P+Q, $$ or $QP+PQ\leq0$. If we conjugate this inequality with $Q$, we get $QPQ+QPQ\leq0$. But $QPQ\geq0$, so $QPQ=0$. Then $$ 0=QPQ=(PQ)^*PQ, $$ and then $PQ=0$. By taking adjoints, $QP=0$.

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2. If $PQ=0$, it follows by taking adjoints that $QP=0$. And $$ (P+Q)^2=P^2+Q^2+QP+PQ=P+Q. $$

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3. If $(P+Q)^2=P+Q$, then by the C$^*$-identity $$ \|P+Q\|^2=\|(P+Q)^2\|=\|P+Q\|, $$ so either $\|P+Q\|=0$ (which by positivity would force $P=Q=0$) or $\|P+Q\|=1$.

Answer 2

Let's enumerate the conditions: (1) $P\perp P'$; (2) $0=PP'=P'P$; (3) $\Sigma P$ is a projection.

We can assume that $A\subset B(H)$ for some Hilbert space $H$ (this basically follows from the GNS construction); also, remember that for positive operators $T\in B(H)$ we have $\langle T\xi,\xi\rangle\geq 0$ for all $\xi\in H$.

For (1)$\Rightarrow$(2), assume that $\xi\in P(H)$ and note that $\langle (P+P')\xi,\xi\rangle\leq\langle\xi,\xi\rangle=\|\xi\|^2$ by assumption (since $0\leq P+P'\leq 1$). Since $\langle P\xi,\xi\rangle=\|\xi\|^2$, we have $ \|P'\xi\|^2=\langle P'\xi,\xi\rangle=0$, so that $\xi=(1-P')\xi+P'\xi=(1-P')\xi$. Thus for all $\eta\in H$, note that $P\eta\in P(H)$, so that we have $P\eta=(1-P')P\eta$ by what we found above. This means that $P'P\eta=0$ for all $\eta\in H$, so $P'P=0$. Taking the adjoint yields (2). (3) follows from (2) by just computing, and any projection $Q\in A$ satisfies $0\leq Q\leq 1$, so if (3) holds, then $\sigma(\Sigma P)\subset[0,1]$ immediately.