A partial isometry element of a C*-algebra 
is an element 
satisfying 
, where 
is the involution. Equivalently,
is a self-adjoint idempotent.
The element 
is then also a self-adjoint idempotent (Courtney et al. 2024, p. 4).
For a bounded operator 
on a Hilbert space, 
denotes its adjoint operator. The condition says
that 
preserves the norm on the orthogonal
complement of its kernel and is
zero on that kernel. The operators 
and 
are the orthogonal projections onto this
initial subspace and the range
of 
,
respectively (Anantharaman and Popa, pp. 33-34). A polar
decomposition uses such a bounded operator
as its isometric factor.
Factors. Undated draft, pp. 33-34. 
-Algebras. GOALS lecture notes, pp. 4 and 20-21,
2024. 
-Algebras
and Operator Theory. New York: Academic Press, 1990.