  For a braided vector space $(V,\sigma)$ with braiding $\sigma$ of Hecke type,
we introduce three associative algebra structures on the space
$\oplus_{p=0}^{M}\mathrm{End}S_\sigma^p(V)$ of graded endomorphisms of the
quantum symmetric algebra $S_\sigma(V)$. We use the second product to construct
a new trace. This trace is an algebra morphism with respect to the third
product. In particular, when $V$ is the fundamental representation of
$\mathcal{U}_{q}\mathfrak{sl}_{N+1}$ and $\sigma$ is the action of the
$R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$.
