One-parameter isometry groups and inclusions between operator algebras

TL;DR

This paper studies one-parameter isometry groups on Banach spaces and operator algebras, exploring their generators, applications to quantum groups, and density results in the context of algebra inclusions.

Contribution

It provides a unified analysis of generators of one-parameter isometry groups, applies these to operator algebra inclusions, and advances understanding of their structural properties.

Findings

Unified the notion of generators across different constructions.

Applied the theory to quantum group scaling automorphisms.

Established a Kaplansky Density type result for operator algebra graphs.

Abstract

We make a careful study of one-parameter isometry groups on Banach spaces, and their associated analytic generators, as first studied by Cioranescu and Zsido. We pay particular attention to various, subtly different, constructions which have appeared in the literature, and check that all give the same notion of generator. We give an exposition of the "smearing" technique, checking that ideas of Masuda, Nakagami and Woronowicz hold also in the weak$^*$-setting. We are primarily interested in the case of one-parameter automorphism groups of operator algebras, and we present many applications of the machinery, making the argument that taking a structured, abstract approach can pay dividends. A motivating example is the scaling group of a locally compact quantum group $\mathbb G$ and the fact that the inclusion $C_0(\mathbb G) \rightarrow L^\infty(\mathbb G)$ intertwines the relevant scaling groups. Under this general setup, of an inclusion of a $C^*$-algebra into a von Neumann algebra intertwining automorphism groups, we show that the graphs of the analytic generators, despite being only non-self-adjoint operator algebras, satisfy a Kaplansky Density style result. The dual picture is the inclusion $L^1(\mathbb G)\rightarrow M(\mathbb G)$, and we prove an "automatic normality" result under this general setup. The Kaplansky Density result proves more elusive, as does a general study of quotient spaces, but we make progress under additional hypotheses.