  Main result: If a C*-algebra is simple, $\sigma$-unital, has finitely many
extremal traces, and has strict comparison of positive elements by traces, then
its multiplier also has strict comparison of positive elements by traces. The
same results holds if "finitely many extremal traces" is replaced by
"quasicontinuous scale".
  A key ingredient in the proof is that every positive element in the
multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be
approximated by a bi-diagonal series.
  An application of strict comparison: If the algebra is a simple separable
stable $\sigma$-unital with real rank zero, stable rank one, strict comparison
of positive elements by traces, then whether a positive element is a linear
combination of projections depends on the trace values of its range projection.
