The (fermionic) Fock space provides a general model in quantum mechanics to construct the quantum states of an unknown number of identical particles from a single particle Hilbert space $V$. The Fock space $\mathfrak{F}$ is (the Hilbert space completion of) the direct sum of the (anti-)symmetric tensors in the (anti-)symmetric tensor powers of $V$ called infinite wedge space. Certain Lie algebras of infinite dimension act on $\mathfrak{F}$ where their characters play a crucial role in representation theory and quantum computations. One can consider a generalization of Fock space namely the Fock space of level $l$ denoted by $\mathfrak{F}^l$ realized as the fermionic operators on $V=(\mathbb{C}^l \otimes \mathbb{C}^{\infty}) \bigoplus (\mathbb{C}^{l*} \otimes \mathbb{C}^{{\infty}*})$ where $\mathbb{C}^{\infty}$ is a vector space with basis $w_r, r \in -1/2-\mathbb{Z}_+$, with the dual indexed by $w_{-r}$, and $\mathbb{C}^l$ has basis $v^{+,i}$ with dual $v^{-,i}$. In this case the fermionic operators are written as $\psi_r^{\pm,i}=v^{\pm,i} \otimes w_r$. The Fock space $\mathfrak{F}^l$ are representations of $\widehat{\mathfrak{gl}_{\infty}} \otimes GL_l \mathbb{C}$. The former case corresponds to $l=1$. There are duality (decompositions) formulas of the Fock space of level $l$, into irreducible representations that are highest weight representations of $\mathfrak{gl}_{\infty} \otimes GL_l \mathbb{C}$.

We define a generalization of the Fock space into $F^{\infty}$ and then express a duality of Howe type for pair of Lie algebras $(\widehat{gl}_{\infty} , \mathfrak{gl}_{\infty})$ that decomposes $F^{\infty}$ into irreducible representations. This enables us to compute the tarce of a

vertex operator with infinitely many Casimirs. The computation uses a calculation of the trace in a work by Bloch-Okounkov on the character of the infinite

wedge representation.