  By classical results of Herglotz and F. Riesz, any bounded analytic function
in the complex unit disk has a unique inner-outer factorization. Here, a
bounded analytic function is called \emph{inner} or \emph{outer} if
multiplication by this function defines an isometry or has dense range,
respectively, as a linear operator on the Hardy Space, $H^2$, of analytic
functions in the complex unit disk with square-summable Taylor series. This
factorization can be further refined; any inner function $\theta$ decomposes
uniquely as the product of a \emph{Blaschke inner} function and a
\emph{singular inner} function, where the Blaschke inner contains all the
vanishing information of $\theta$, and the singular inner factor has no zeroes
in the unit disk.
  We prove an exact analog of this factorization in the context of the full
Fock space, identified as the \emph{Non-commutative Hardy Space} of analytic
functions defined in a certain multi-variable non-commutative open unit disk.
