  A trace on the C^*-algebra A of quasi-local operators on an open manifold is
described, based on the results in \cite{RoeOpen}. It allows a description `a
la Novikov-Shubin \cite{NS2} of the low frequency behavior of the
Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms
of such a trace is proved to coincide with a metric invariant, which we call
asymptotic dimension, thus giving a large scale ``Weyl asymptotics'' relation.
Moreover, in analogy with the Connes-Wodzicki result \cite{CoCMP,Co,Wo}, the
asymptotic dimension d measures the singular traceability (at 0) of the
Laplace-Beltrami operator, namely we may construct a (type II_1) singular trace
which is finite on the $^*$-bimodule over A generated by $\Delta^{-d/2}$.
