  It is known that the multiplier algebra of an approximately unital and
nondegenerate $L^p$-operator algebra is again an $L^p$-operator algebra. In
this paper we investigate examples that drop both hypotheses. In particular, we
show that the multiplier algebra of $T_2^p$, the algebra of strictly upper
triangular $2 \times 2$ matrices acting on $\ell_2^p$, is still an
$L^p$-operator algebra for any $p$. To contrast this result, we first provide a
thorough study of the augmentation ideal of $\ell^1(G)$ for a discrete group
$G$. We use this ideal to define a family of non approximately unital
degenerate $L^p$-operator algebras, $F_{r,0}^p(\Bbb{Z}/3\Bbb{Z})$, whose
multiplier algebras cannot be represented on any $L^q$-space for any $q \in [1,
\infty)$ as long as $p \in [1, p_0] \cup [p_0', \infty)$, where $p_0=1.606$ and
$p_0'$ is its H\"older conjugate.
