$L^p$-Spaces as Quasi *-Algebras

TL;DR

This paper explores the structure of $L^p$ spaces as quasi *-algebras over $C(X)$, demonstrating *-semisimplicity for $p \, \geq \, 2$ and discussing related implications.

Contribution

It introduces $L^p$-spaces as CQ*-algebras over $C(X)$ and proves their *-semisimplicity for $p \, \geq \, 2$, extending algebraic understanding.

Findings

$L^p$ spaces can be structured as CQ*-algebras over $C(X)$.

For $p \, \geq \, 2$, these structures are *-semisimple.

The paper discusses consequences of *-semisimplicity in this context.

Abstract

The Banach space $L^p(X,\mu)$, for $X$ a compact Hausdorff measure space, is considered as a special kind of quasi *-algebra (called CQ*-algebra) over the C*-algebra $C(X)$ of continuous functions on $X$. It is shown that, for $p \geq 2$, $(L^p(X,\mu), C(X))$ is *-semisimple (in a generalized sense). Some consequences of this fact are derived.