C-star algebra ($C^{\ast }$-algebra)

A C-star algebra is a complex Banach algebra with an involution satisfying $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

An element $a$ of a C-star algebra is positive when $a=b^{\ast }b$ for some $b$, equivalently when $a=a^{\ast }$ and $\sigma (a)\subseteq [0,\infty )$. Continuous functional calculus gives it a unique positive square root $a^{1/2}$.

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.

Every bounded operator on a Hilbert space has a polar decomposition $T=U|T|$, where $|T|=(T^{\ast }T{)}^{1/2}$ and $U$ is a partial isometry with $\ker U=\ker T$. On $\mathrm{im}|T|$, it is defined by $U(|T|x)=Tx$.