Partial isometry

An operator $U$ on a Hilbert space is a partial isometry when it is isometric on $(\ker U{)}^{\perp }$. Equivalently, $U^{\ast }U$ is the orthogonal projection onto this initial space.

For a nonzero partial isometry $u$ in a C-star algebra, put $p=u{u}^{\ast }$ and $q=u^{\ast }u$. The closed linear space $pAq$ becomes a unital C-star algebra with the displayed operations and inherited norm, with unit $u$. The map $a\mapsto u^{\ast }a$ is an isometric -isomorphism onto the ordinary corner $qAq$, with inverse $b\mapsto ub$. The identities $u{u}^{\ast }a=a$ and $u^{\ast }ub=b$ verify the inverse maps, while contractivity of both multiplications gives isometry. Products and involutions transfer directly, giving all algebra and C-norm axioms. This construction turns a rectangular operator corner into a square algebra by using the partial isometry as its unit.