  The framework of the paper is that of the full Fock space ${\Cal F}^2({\Cal
H}_n)$ and the Banach algebra $F^\infty$ which can be viewed as non-commutative
analogues of the Hardy spaces $H^2$ and $H^\infty$ respectively.
  An inner-outer factorization for any element in ${\Cal F}^2({\Cal H}_n)$ as
well as characterization of invertible elements in $F^\infty$ are obtained. We
also give a complete characterization of invariant subspaces for the left
creation operators $S_1,\cdots, S_n$ of ${\Cal F}^2({\Cal H}_n)$. This enables
us to show that every weakly (strongly) closed unital subalgebra of
$\{\varphi(S_1,\cdots,S_n):\varphi\in F^\infty\}$ is reflexive, extending in
this way the classical result of Sarason [S].
  Some properties of inner and outer functions and many examples are also
considered.
