  We study a family of representations of the canonical commutation relations
(CCR)-algebra (an infinite number of degrees of freedom), which we call
admissible. The family of admissible representations includes the Fock-vacuum
representation. We show that, to every admissible representation, there is an
associated Gaussian stochastic calculus, and we point out that the case of the
Fock-vacuum CCR-representation in a natural way yields the operators of
Malliavin calculus. And we thus get the operators of Malliavin's calculus of
variation from a more algebraic approach than is common. And we obtain explicit
and natural formulas, and rules, for the operators of stochastic calculus. Our
approach makes use of a notion of symmetric (closable) pairs of operators. The
Fock-vacuum representation yields a maximal symmetric pair. This duality
viewpoint has the further advantage that issues with unbounded operators and
dense domains can be resolved much easier than what is possible with
alternative tools. With the use of CCR representation theory, we also obtain,
as a byproduct, a number of new results in multi-variable operator theory which
we feel are of independent interest.
