\abstract Let $\Cal A\subseteq\Cal B(H)$ be a \cstar\ of operators
and let $P_1\leq P_2\leq\dots$ be an increasing sequence of
finite dimensional projections in $\Cal B(H)$.  
In a previous paper \cite{3} we developed methods for computing
the spectrum of self adjoint operators $T\in\Cal A$ in terms of
the spectra of the associated sequence of finite dimensional 
compressions $P_nTP_n$.  In a suitable context, we
showed that this is possible when $P_n$ increases to $\I$.  In this
paper we drop that hypothesis and obtain an appropriate
generalization of the main results of \cite{3}.  

Let $P_+=\lim_nP_n$, $H_+=P_+H$.  The set $\Cal A_+\subseteq\Cal B(H_+)$ 
of all compact perturbations of operators $P_+T\!\restriction_{H_+}$, 
$T\in\Cal A$, is a
\cstar\ which is somewhat analogous to the Toeplitz \cstar\ acting 
on $H^2$.
Indeed, in the most important examples $\Cal A$ is a simple unital
\cstar\ having a unique tracial state, the operators in $\Cal A$ are 
``bilateral", those in $\Cal A_+$ are ``unilateral", and there is a
short exact sequence of \cstar s
$$
0\to\Cal K\to\Cal A_+\to\Cal A\to 0
$$
whose features are central to this problem of approximating spectra
of operators in $\Cal A$ in terms of the eigenvalues of their finite 
dimensional compressions along the given filtration.  

This work was undertaken in order to develop an efficient method
for computing the spectra of discretized Hamiltonians of one
dimensional quantum
systems in terms of ``unilateral" tridiagonal $n\times n$ matrices.  
The solution of that problem is presented in Theorem 3.4.  
\endabstract