Multiplier algebras of $L^p$-operator algebras

Andrey Blinov, Alonso Delfín and Ellen Weld

Vol. 333 (2024), No. 2, 197–227

DOI: 10.2140/pjm.2024.333.197

Abstract

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 nonapproximately unital degenerate $L^p$-operator algebras, $F_0^p(\mathbb{Z}∕3\mathbb{Z})$, whose multiplier algebras cannot be represented on any $L^q$-space for any $q\in [1,\infty )$ as long as $p\in [1,p_0]\cup [p_0^{\prime },\infty )$, where $p_0=1.606$ and $p_0^{\prime }$ is its Hölder conjugate.

Keywords

multiplier algebra, augmentation ideal, representations on $L^p$-spaces, $L^p$-group algebras, $p$-operator norms

Mathematical Subject Classification

Primary: 46H15, 46H35

Secondary: 47L10

Milestones

Received: 25 March 2024

Revised: 12 November 2024

Accepted: 30 November 2024

Published: 28 December 2024