Multiplier algebras of $L^p$-operator algebras

Andrey Blinov, Alonso Delfín, Ellen Weld

Published 2024-03-24Version 1

We show that the multiplier algebra of an approximately unital and nondegenerate $L^p$-operator algebra is again an $L^p$-operator algebra. We then 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 provide a thorough study of the $L^1$-operator algebra $\ell_0^1(G)$, the augmentation ideal of $\ell^1(G)$ for a discrete group $G$, and then show that, at least when $G=\mathbb{Z}/3\mathbb{Z}$, its multiplier algebra is not an $L^p$-operator algebra for any $p \in [1,\infty)$.