  We define a complete set of supertraces on the algebra $SH_N(\nu)$, the
algebra of observables of the $N$-body rational Calogero model with harmonic
interaction. This result extends the previously known results for the simplest
cases of $N=1$ and $N=2$ to arbitrary $N$. It is shown that $SH_N(\nu)$ admits
$q(N)$ independent supertraces where $q(N)$ is a number of partitions of $N$
into a sum of odd positive integers, so that $q(N)>1$ for $N\ge 3$. Some
consequences of the existence of several independent supertraces of $SH_N (\nu
)$ are discussed such as the existence of ideals in associated $W_{\infty}$ -
type Lie superalgebras.
