subalgebra of $\BB(\FFF)$ by $\pi(  \mathfrak{A}(\HH) )$.

The Fock representation of $\mathfrak{A}(\HH)$ on $\FFF(\HH)$ is irreducible, i.e., the commutant of $\pi(  \mathfrak{A}(\HH) )$ in $\BB(\FFF)$ is trivial. Therefore the bi-commutant of $\pi(  \mathfrak{A}(\HH) )$, in which it is weakly dense, is all of $\BB(\FFF)$.

Given an ordered orthonormal basis $(h_1,h_2,\ldots)$ of $\HH$, one can build an orthonormal basis of $\FFF(\HH)$, called a “Fock basis" or “occupation number" basis, using the orbitals $h_i$ as “reference" orbitals. The Fock basis vectors represent configurations of particles in the reference orbitals and are indexed by “occupation lists" $$\bn = \big(\bn(1),\bn(2),\bn(3),\ldots \big)$$ such that $\sum \bn(i) < \infty$, that is, such that the total number of particles in the configuration is finite. The set $$\NN \equals \Big\{ \big(\bn(1),\bn(2),\bn(3),\ldots \big) :  \bn(i) \in \{ 0,1\},\  \sum \bn(i) < \infty \Big\}$$ indexes the possible configurations of fermions in the modes $(h_1,h_2,\ldots)$. The occupation list ${\bf 0} = (0,0,0,\ldots)$ is the index of the vacuum vector $|\Omega \rangle$, i.e., $| {\bf 0}\rangle = |\Omega \rangle$. For $\bn \in \NN$ with $\sum \bn(i) > 0$, define the vector $$\label{Fock basis vector}
         |\bn\rangle \equals \hat{a}^*(h_1)^{\bn(1)}\hat{a}^*(h_2)^{\bn(2)} \cdots |\Omega \rangle$$ (since the exponents $\bn(i)$ are eventually $0$, only finitely many creators appear 