A well-known theorem of Sarason [11] asserts that if
$\left[ {{T}_{f}},\,{{T}_{h}} \right]$
is compact for every
$h\,\in \,{{H}^{\infty }}$
, then
$f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$
. Using local analysis in the full Toeplitz algebra
$T\,=\,T\left( {{L}^{\infty }} \right)$
, we show that the membership
$f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$
can be inferred from the compactness of a much smaller collection of commutators
$\left[ {{T}_{f}},\,{{T}_{h}} \right]$
. Using this strengthened result and a theorem of Davidson [2], we construct a proper
${{C}^{*}}$
-subalgebra
$T\left( \mathcal{L} \right)$
of
$T$
which has the same essential commutant as that of
$T$
. Thus the image of
$T\left( \mathcal{L} \right)$
in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra
$\mathcal{S}$
of
$T$
is capable of conferring the membership
$f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$
through the compactness of the commutators
$\left\{ \left[ {{T}_{f,}}\,S \right]\,:\,S\,\in \,\mathcal{S} \right\}$
.