  We present an extension of the trace of a vertex operator and explain a
representation-theoretic interpretation of the trace. Specifically, we consider
a twist of the vertex operator with infinitely many Casimir operators and
compute its trace as a character formula. To do this, we define the Fock space
of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality between
$\mathfrak{gl}_{\infty}$ and
$\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which
provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible
representations with joint highest weight vector for $\mathfrak{gl}_{\infty}$
and $\mathfrak{a}_{\infty}$. The decomposition of the Fock space
$\mathfrak{F}^{\infty}$ into highest weight representations provides a method
to calculate and interpret the extended trace.
