If
$A$
is a
$\sigma $
-unital
${{C}^{*}}$
-algebra and
$a$
is a strictly positive element of
$A$
, then for every compact subset
$K$
of the complete regularization Glimm
$(A)$
of Prim
$(A)$
there exists
$\alpha \,>\,0$
such that
$K\,\subset \,\{G\,\in \,\text{Glimm(}A\text{)}\,\text{ }\!\!|\!\!\text{ }\,\left\| a\,+\,G \right\|\,\ge \,\alpha \}$
. This extends a result of J. Dauns to all
$\sigma $
-unital
${{C}^{*}}$
-algebras. However, there exist a
${{C}^{*}}$
-algebra
$A$
and a compact subset of Glimm
$(A)$
that is not contained in any set of the form
$\{G\,\in \,\text{Glimm(}A\text{)}\,\text{ }\!\!|\!\!\text{ }\,\left\| a+\,G \right\|\,\ge \,\alpha \},\,a\in \,A$
and
$\alpha \,>\,0$
.