Projection in a C-star algebra ($p=p^{\ast }=p^2$)

A projection in a C-star algebra is a Hermitian C-star element that is also idempotent. Its spectrum is contained in $\{0,1\}$, and a nonzero projection has norm one by the C-star identity. If $u$ is a partial isometry, $u^{\ast }u$ and $u{u}^{\ast }$ are its initial and final projections. For a projection $p$, the closed subalgebra $pAp$ has unit $p$ and the inherited norm and involution; it is the corresponding corner C-star algebra. In an operator representation, such an element acts as an orthogonal projection.