  The completion of a (normed) $C^*$-algebra $A_0[\| \cdot \|_0]$ with respect
to a locally convex topology $\tau$ on $A_0$ that makes the multiplication of
$A_0$ separately continuous is, in general, a quasi *-algebra, and not a
locally convex *-algebra. In this way, one is led to consideration of locally
convex quasi $C^*$-algebras, which generalize $C^*$-algebras in the context of
quasi *-algebras. Examples are given and the structure of these relatives of
$C^*$-algebras is investigated.
