  In this paper we examine in detail the non-commutative symplectic algebra
underlying quantum dynamics. We show that this algebra contains both the
Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner
distribution as the kernel of the density matrix. The underlying
non-commutative geometry can be projected into either of two Abelian spaces,
so-called `shadow phase spaces'. One of these is the phase space of Bohmian
mechanics, showing that it is a fragment of the basic underlying algebra. The
algebraic approach is much richer, giving rise to two fundamental dynamical
time development equations which reduce to the Liouville equation and the
Hamilton-Jacobi equation in the classical limit. They also include the
Schr\"{o}dinger equation and its wave function, showing that these features are
a partial aspect of the more general non-commutative structure. We discuss
briefly the properties of this more general mathematical background from which
the non-commutative symplectic algebra emerges.
