Multiplier algebras of $L^p$-operator algebras

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 ${\mathrm{\ell }}_2^p$, 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 ${\mathrm{\ell }}^1(G)$ for a discrete group $G$. We use this ideal to define a family of nonapproximately unital degenerate $L^p$-operator algebras, $F_0^p(\mathbb{Z}\mathrm{/}3\mathbb{Z})$, whose multiplier algebras cannot be represented on any $L^q$-space for any $q\in [1,\mathrm{\infty })$ as long as $p\in [1,p_0]\cup [p_0^{\mathrm{\prime }},\mathrm{\infty })$, where $p_0=1.606$ and $p_0^{\mathrm{\prime }}$ is its Hölder conjugate.