## Properties of projections in a C*-algebra

A *projection* in a C*-algebra is an element $p$ satisfying $$p=p^*=p^2.$$
(The above equation is equivalent to $p=p^*p$, noting that $p^*=(p^*p)^*=p^*p=p$ and thus $p=p^*p=p^2$.)

**Property 1**. Let $A$ be a C*-algebra, $p$ and $q$ are two projections in $A$. Then $$p\leq q\Leftrightarrow pq=p\Leftrightarrow qp=p.$$

*Proof*. $p \leq q\Rightarrow pq=p$:

Suppose $a=q-p\geq 0$, then \begin{align*}
pq-p=pa,
\end{align*}
hence \begin{align*}
p(1-q)p=-pap,
\end{align*}
Thus $p(1-q)p=0$, $\|p-pq\|^2=\|p(1-q)p\|=0$ and $p=pq$.

**Property 2**.
$$pq=0\Leftrightarrow p\circ q=0$$ where $p\circ q=\frac{pq+qp}{2}$.

*Proof*. If $p\circ q=0$, then $$pq=-qp,$$ hence \begin{align*}pqp=p(pq)p=p(-qp)p=-pqp.\end{align*} So, $pqp=0$, $\|pq\|^2=\|pqp\|=0$, $pq=0$.

## the Relationship between Projections and Unitaries

Update. We can use functional calculus to obtain unitaries or projections.

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If $u$ is a unitary, then $\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & u^*\\ u &1 \end{array}\right)$ is a projection;

If $p$ is a projection, then $\left(\begin{array}{cc} p & 1-p\\ 1-p & p \end{array}\right)$ is a unitary.