I am trying to show that if A is an arbitrary C*-algebra, $p$ is a projection if and only if $p^{*}p = p$. Now since the C*-algebra is arbitrary, we only have a norm and not necessarily an inner product, and I do not know how I can use the adjoint in this setting. Can someone give me a hint?
2026-03-25 19:00:34.1774465234
Projection on C*-Algebra
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The identity $p^*p=p$ implies $$p^*=(p^*p)^*=p^*p=p,$$ and thus $$p=p^*p=p^2$$