Multiplier algebras of $L^p$-operator algebras
Andrey Blinov, Alonso Delfín, Ellen Weld
TL;DR
The paper investigates when multiplier algebras of $L^p$-operator algebras remain within the same analytic class after removing the usual hypotheses of approximate unitality or nondegenerate representation. It presents two contrasting constructions: the multiplier algebra of the degenerate algebra $T_2^p$ is still an $L^p$-operator algebra, while the multiplier algebra of the augmentation-ideal construction $F_0^p(\mathbb{Z}/3\mathbb{Z})$ can fail to admit any $L^q$-operator representation for a broad range of $p$ values. A detailed analysis of augmentation ideals for finite groups shows these algebras have no cai and cannot be nondegenerately represented, while the Gelfand-transform structure for $\ell_0^1(G)$ in finite abelian groups yields a concrete idempotent description for their multiplier algebras. The work also underscores the sensitivity of $L^p$-operator algebra norms by demonstrating norms on $\mathbb{C}^n$ that either support or preclude $L^p$-operator representations, with open questions about full norm classification on $\mathbb{C}^n$ in this context.
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, $F_{0}^p(\Bbb{Z}/3\Bbb{Z})$, 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 Hölder conjugate.