  This paper studies the algebraic aspect of a general abelian coset theory
with a work of Dong and Lepowsky as our main motivation. It is proved that the
vacuum space $\Omega_{V}$ (or the space of highest weight vectors) of a
Heisenberg algebra in a general vertex operator algebra $V$ has a natural
generalized vertex algebra structure in the sense of Dong and Lepowsky and that
the vacuum space $\Omega_{W}$ of a $V$-module $W$ is a natural
$\Omega_{V}$-module. The automorphism group $\Aut_{\Omega_{V}}\Omega_{V}$ of
the adjoint $\Omega_{V}$-module is studied and it is proved to be a central
extension of a certain torsion free abelian group by $\C^{\times}$. For certain
subgroups $A$ of $\Aut_{\Omega_{V}}\Omega_{V}$, certain quotient algebras
$\Omega_{V}^{A}$ of $\Omega_{V}$ are constructed. Furthermore, certain functors
among the category of $V$-modules, the category of $\Omega_{V}$-modules and the
category of $\Omega_{V}^{A}$-modules are constructed and irreducible
$\Omega_{V}$-modules and $\Omega_{V}^{A}$-modules are classified in terms of
irreducible $V$-modules. If the category of $V$-modules is semisimple, then it
is proved that the category of $\Omega_{V}^{A}$-modules is semisimple.
