Description:
<p>Let <italic>A</italic> be a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msup>
<mml:mi>C</mml:mi>
<mml:mo>ā<!-- ā --></mml:mo>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{C^ \ast }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-algebra which either allows a faithful separable representation or is postliminal. We prove that <italic>A</italic> then admits a smallest faithful representation if and only if the ideal of compact elements is an essential ideal in <italic>A</italic>.</p>